KAM Theory:quasi-periodicity in dynamical systems

Henk W. BroerDepartment of Mathematics and Computing Science,

University of Groningen,Blauwborgje 3, 9747 AC Groningen, The Netherlands

E-mail: broer@math.rug.nl

Mikhail B. SevryukInstitute of Energy Problems of Chemical Physics,

The Russia Academy of Sciences,Leninskiı prospect 38, Bldg. 2, Moscow 119334, Russia

E-mail: sevryuk@mccme.ru

August 13, 2007

Abstract

We present Kolmogorov–Arnold–Moser (orKAM ) Theory regarding typical-ity of quasi-periodic invariant tori, partly from a historical and partly from apedagogical point of view. At the same time we aim at a unified approach ofthe theory in various dynamical settings: the ‘classical’ Hamiltonian settingof Lagrangean tori, the Hamiltonian lower dimensional isotropic tori, thedissipative case of quasi-periodic attractors, etc. Also we sketch the theoryof quasi-periodic bifurcations, where resonances cause Cantorization andfraying of the bifurcation sets known from the cases of equilibrium pointsand periodic orbits. Here the concept of Whitney differentiability plays acentral role, which locally organizes the nowhere dense union of persistentquasi-periodic invariant tori, of positive measure. At thelevel of torus bun-dles this Cantorization is observed as well, where the geometry of the torusbundles turns out to be persistent. In the meantime we brieflydeal with thenatural affine structure of quasi-periodic tori, with uniqueness of most ofthe KAM tori, and with the mechanisms of the destruction of resonantun-perturbed tori. Other parts of the theory, such as the Hamiltonian higher

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dimensional coisotropic tori, the so-called atropic tori,and the excitation ofelliptic normal modes of lower dimensional tori, are also discussed.

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Contents

1 Introduction 51.1 The ‘classical’KAM theorem . . . . . . . . . . . . . . . . . . . . 51.2 Related developments: outline . . . . . . . . . . . . . . . . . . . 61.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Complex linearization 92.1 Formal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Convergence and small divisors . . . . . . . . . . . . . . . . . . 102.3 Measure and category . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 KAM Theory for circle and annulus maps 123.1 Circle maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Small divisors again . . . . . . . . . . . . . . . . . . . . 133.1.2 AKAM theorem for circle maps . . . . . . . . . . . . . . 143.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Area preserving annulus maps . . . . . . . . . . . . . . . . . . . 203.2.1 Moser’s Twist Mapping Theorem . . . . . . . . . . . . . 203.2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 KAM Theory for flows 234.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Affine structure . . . . . . . . . . . . . . . . . . . . . . . 254.1.2 The perturbation problem . . . . . . . . . . . . . . . . . 25

4.2 Families of normally hyperbolic quasi-periodic tori . .. . . . . . 284.2.1 Formulation of the normally hyperbolicKAM theorem . . 284.2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 KAM Theory for Lagrangean tori in Hamiltonian systems . . . . . 314.3.1 Formulation of the LagrangeanKAM theorem . . . . . . . 314.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4 Applications of the LagrangeanKAM Theorem 6 . . . . . . . . . 364.4.1 Applications in Classical, Quantum, and StatisticalMe-

chanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Further developments inKAM Theory 395.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1.1 Unicity ofKAM tori . . . . . . . . . . . . . . . . . . . . . 405.1.2 Paley–Wiener estimates and Diophantine frequencies. . . 41

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5.2 ParametrizedKAM Theory . . . . . . . . . . . . . . . . . . . . . 435.2.1 The parametrized dissipativeKAM theorem . . . . . . . . 445.2.2 Direct consequences of the parametrized approach . . .. 485.2.3 Reducibility issues . . . . . . . . . . . . . . . . . . . . . 49

6 Quasi-periodic bifurcations: dissipative setting 506.1 Quasi-periodic Hopf bifurcation . . . . . . . . . . . . . . . . . . 52

6.1.1 Persistent quasi-periodicn-tori . . . . . . . . . . . . . . . 536.1.2 Fattening the parameter domain of invariantn-tori . . . . 546.1.3 The parameter domain of invariant(n + 1)-tori . . . . . . 55

6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2.1 Fraying . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2.2 Non-parallel dynamics . . . . . . . . . . . . . . . . . . . 576.2.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . 58

7 Quasi-periodic bifurcation theory in other settings 607.1 Hamiltonian cases . . . . . . . . . . . . . . . . . . . . . . . . . . 607.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

8 Further Hamiltonian KAM Theory 628.1 Exponential condensation . . . . . . . . . . . . . . . . . . . . . . 638.2 Destruction of resonant tori . . . . . . . . . . . . . . . . . . . . . 658.3 Lower dimensional isotropic invariant tori . . . . . . . . . .. . . 67

8.3.1 The parametrized HamiltonianKAM theorem . . . . . . . 688.3.2 Lower dimensional tori in individual Hamiltonian systems 738.3.3 Historical remarks . . . . . . . . . . . . . . . . . . . . . 78

8.4 Excitation of elliptic normal modes . . . . . . . . . . . . . . . . .808.5 Higher dimensional coisotropic invariant tori . . . . . . .. . . . 818.6 Atropic invariant tori . . . . . . . . . . . . . . . . . . . . . . . . 83

9 Whitney smooth bundles ofKAM tori 849.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849.2 Formulation of the globalKAM theorem . . . . . . . . . . . . . . 859.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

9.3.1 Example: the spherical pendulum . . . . . . . . . . . . . 889.3.2 Monodromy in the nearly integrable case . . . . . . . . . 89

9.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

10 Conclusion 92

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1 Introduction

Kolmogorov–Arnold–Moser (orKAM ) Theory was developed for conservative(Hamiltonian) dynamical systems that are nearly integrable. Integrable systemsin their phase space contain lots of invariant tori andKAM Theory establishes per-sistence of such tori, which carry quasi-periodic motions.We present this theorywhich begins with Siegel’s and Kolmogorov’s pioneering work in the 1940’s and50’s.

Since Moser’s results from the 1960’s it is known thatKAM Theory extendsoutside the world of Hamiltonian systems. Indeed, as will beexplained below,families of quasi-periodic attractors can be dealt with in the same way as quasi-periodic Lagrangean invariant tori in Hamiltonian systems. In both cases a Kolmogorov-like nondegeneracy condition is needed on the way frequencies vary with the un-perturbed tori. The background is Moser’s Lie algebra version of KAM Theory.There are other types of nondegeneracy conditions as well, for instance, the so-called Russmann condition. All our formulations include Whitney differentiableconjugations with collections of Diophantine quasi-periodic tori in integrable ap-proximations. This part of the theory was initiated by Lazutkin and Poschel in the1970’s and 80’s. From this, for a large class ofKAM tori, uniqueness follows.

A general type of nondegeneracy, involving unfolding parameters and transver-sality, was developed in the late 1980’s by Broer, Huitema, and Takens. It can beshown to encompass (in a sense to be made precise) both the Kolmogorov andRussmann nondegeneracy. Also it is at the basis of the quasi-periodic bifurca-tion theory. It turns out that the standard (semi-algebraic) bifurcation diagrams,as known for equilibria and periodic solutions, in the quasi-periodic setting occurin a ‘Cantorized’ and, sometimes, ‘frayed’ way. These developments took placeduring the 1990’s and round the turn of the century.

Recently a globalKAM Theory was constructed, which leads to Whitney smoothbundles of invariant tori.

1.1 The ‘classical’KAM theorem

At the International Congress of Mathematicians in 1954, held in Amsterdam,A.N. Kolmogorov gave a closing lecture with the title “The general theory ofdynamical systems and classical mechanics” [239]. Among many other thingshe discussed his paper [238]. The event took place in the Amsterdam Concert-gebouw and it has played a major role in the developments of the DynamicalSystem Theory and of Mathematical Physics, in particular ofwhat is now calledKolmogorov–Arnold–Moser (orKAM ) Theory. We like to note that the term ‘KAM

Theory’ was first used in [226,455].In this lecture Kolmogorov considered the occurrence of multi- or quasi-periodic

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motions, which in the phase space are confined to invariant tori. He restrictedhimself to conservative, or Hamiltonian, dynamical systems, as these are gener-ally used for modelling in classical mechanics. Invariant Lagrangean tori thatcarry quasi-periodic motions were well-known to occur in Liouville integrablesystems, and Kolmogorov’s paper [238] and lecture [239] dealt with the persis-tence of these tori under small, non-integrable perturbations of the Hamiltonian.Due to so-called small divisors, the corresponding perturbation series diverge ona dense set.

In broad terms,KAM Theory states that generically, in small perturbations ofintegrable systems the union of quasi-periodic Lagrangeaninvariant tori has pos-itive Liouville measure both in the phase space and in the energy hypersurfaces.Here two related nondegeneracy conditions come in play, dealing with the wayfrequencies or frequency ratios vary with the tori in the phase space.

“This theorem is often said to be the first and perhaps foremost result of modernnon-linear dynamics of conservative systems” [353], p. 487(compare with [188]).

1.2 Related developments: outline

Already in the 19th century (in fact, even earlier) the problem of small divisorswas met, notably in perturbation series related to a three-body problem [2,17,20,142, 183, 315, 366, 403]. This fact initiated many developments, partly ending upin KAM Theory as reported on here. H. Poincare played a central role in theseearly developments.

Poincare was also one of the founders of the linearization program to which theexample of the next section belongs. This concerns the linearization near a fixedpoint of a holomorphic diffeomorphism of the complex plane and leads to the firstsolution of a small divisor problem by C.L. Siegel [402], compare with [11]. Herewe first meet the theme of ‘measure versus category’ [334] which is so central inKAM Theory.

Next we turn to the dynamics of circle maps, which goes back toArnold [3,11],followed by a discussion on area preserving twist maps [208,309,316,365].

After this the flow case is considered. First we show that in general quasi-periodic invariant tori have a natural affine structure, which in the Liouville in-tegrable Hamiltonian setting coincides with that given by the Liouville–ArnoldTheorem [5, 12, 14, 17, 129, 297]. However, it was already known to J.K. Moser[310–312,315] thatKAM Theory admits a much greater generality than the worldof Hamiltonian systems. This will be illustrated by treating families of quasi-periodic attractors [326, 361], exactly like the ‘classical’ case of Lagrangean in-variant tori in Hamiltonian systems we next describe.

All KAM theorems below are given in the ‘structural stability’ form, where aconjugation is produced between the Diophantine quasi-periodic tori in the inte-

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grable and the nearly integrable cases. In the present setting we speak of ‘quasi-periodic stability’. In the spirit of Poschel [346], theseconjugations are Whitneydifferentiable; also compare with [113,256,257,413,456,457]. This accounts forthe fact that it is a typical property to have a union of quasi-periodic invarianttori of positive Hausdorff measure of the appropriate dimension [67, 68, 123]. Aproperty is called ‘typical’ if it occurs on an open set of thedynamical systems athand. Regarding the topology on the ‘function space’, one may think of the (weak)Whitney topology for differentiable systems [214, 318], orof the compact-opentopology for holomorphic extensions of real analytic systems, compare with [86].Moreover, Whitney differentiability enables us to show uniqueness for a largeclass of the perturbedKAM tori in several situations [83].

We next describe the ParametrizedKAM Theory, inspired by Moser [311,312],where a general nondegeneracy concept has been developed inthe late 1980’sby Broer, Huitema, and Takens. This BHT nondegeneracy involves a certain(uni)versality of parametrized systems [67–69, 223]. By considering the geom-etry and the number theory of the (Diophantine) quasi-periodic frequency vec-tors, this notion can be shown to encompass both the Kolmogorov and Russmannnondegeneracies [238, 239, 368, 370, 373] (in the latter case, the number of pa-rameters can be drastically reduced). This theory is developed in a structure pre-serving way, usingKAM Theory as formulated for certain Lie algebras of vectorfields [62, 69, 216, 223, 312]. In many Hamiltonian and reversible settings, allparameters can be ‘compensated’ by phase space variables.

The BHT nondegeneracy also is at the basis of the quasi-periodic bifurca-tion theory [35, 69]. It turns out that the standard (semi-algebraic) bifurcationdiagrams in the product of the phase space and the parameter space, as knownfor equilibria and periodic solutions since Whitney, Thom,Mather, and Arnold[11,13,15,16,418], occur in the quasi-periodic setting ina ‘Cantorized’ way: nearthe dense set of resonancesKAM Theory does not work. To show this we dealwith the quasi-periodic Hopf bifurcation in some detail, also indicating certainHamiltonian and reversible analogues [48,49,54,56,57,59,60,62,195–199,216].Here the ‘conventional’ Hopf bifurcation [218] (or the Poincare–Andronov phe-nomenon [11]) for equilibrium points plays a central role.

We next dwell upon other branches ofKAM Theory, confining ourselves withthe Hamiltonian case. The central theme here is quasi-periodic invariant toriwhose dimension is not equal to the numbern of degrees of freedom. Lower di-mensional (of dimensions< n) isotropic tori studied first by Melnikov [302,303]and Moser [311, 312] have been explored in great detail by now, which enablesus to formulate the correspondingKAM theorems on the torus persistence in the‘structural stability’ form. Another type of theorems on lower dimensional toriconcerns families of tori of dimensions` + 1, ` + 2, . . . , n around -tori with par-tially ‘elliptic’ normal behavior, and we consider this topic (the so-called ‘excita-

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tion of elliptic normal modes’) as well. Higher dimensional(of dimensions> n)coisotropic tori introduced intoKAM Theory by Parasyuk [337–340] have beenunderstood much worse than lower dimensional tori, and we present here just areview without precise statements. Very recently, Huang, Cong, and Li [221,222]started examining quasi-periodic invariant tori which are‘atropic’, i.e., neitherisotropic nor coisotropic. The dimension of such atropic tori can be smaller than,equal to, or greater than the number of degrees of freedom. Apart from lowerdimensional isotropic, higher dimensional coisotropic, and atropic tori, we alsoreturn to the ‘classical’ case of Lagrangean invariantKAM tori and describe their‘exponential condensation’ and ‘superexponential stickiness’ in the analytic cate-gory [307] as well as discuss the destruction of resonant unperturbed Lagrangeantori into finite collections of lower dimensional tori (the phenomenon first inves-tigated in detail by Treshchev [419]).

We end with the description of the globalKAM Theory [51] in the ‘classi-cal’ Hamiltonian setting, as this leads to Whitney smooth bundles of invarianttori that inherit the corresponding geometry of the integrable bundle, involvingmonodromy, Chern classes, etc. Here the conjugations of theLagrangeanKAM

theorem are glued together with a Partition of Unity [214,318,407].

1.3 Discussion

The Kolmogorov and BHT nondegeneracies are formulated in local terms thatby the Inverse Function Theorem give rise to open domains in the product of thephase space and parameter space on which certain frequency maps are submer-sions. Most of the results have been formulated in terms of these open domains,which in applications may turn out to be quite large.

Below KAM Theory is developed mostly for Hamiltonian systems (which isjustified partly by historical reasons and partly by reasonsof applications), whereasparallel results exist for other classes of dynamical systems as well. In particular,the reversibleKAM Theory (starting with Moser’s paper [310]) is to a great extentparallel to the Hamiltonian one, see e.g. [10, 18, 22, 48, 49,66–68, 312, 315, 346,380,383,388,395,400,401] (in the case of reversible diffeomorphisms, however,some special effects are exhibited [351]), and the weakly reversibleKAM The-ory has been developed in [18, 380].1 The present set-up is even more general,

1A vector fieldX is said to bereversiblewith respect to a phase space diffeomorphismG ifG conjugatesX to −X , that isG∗X = −X . This means thatG

(a(−t)

)is a solution of the

corresponding systema = X(a) of ordinary differential equations whenevera(t) is. The classicalexample is the Newtonian equations of motionu = F (u), u ∈ RN , which can be written in theform u = v, v = F (u); hereG : (u, v) 7→ (u,−v). Similarly, a diffeomorphismA of a certainmanifold is said to bereversiblewith respect to another diffeomorphismG of the same manifoldif GAG−1 = A−1. Following Arnold’s note [10], one often speaks ofweaklyreversible systems

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also including the class of ‘dissipative’ systems, where nostructure has to be pre-served. As already mentioned, in these cases often parameters are needed for thepersistence of quasi-periodic tori. For earlier, partial overviews ofKAM Theory inthe same general spirit, see [46, 47, 67–69, 312, 388]. We note that, although thepresent theory is mostly being developed for flows, a completely analogous ap-proach exists for diffeomorphisms. Also we like to mention that our bibliographyis partly complementary to those of [17,68,155,199,274,349,399].

For a first acquaintance withKAM Theory, the introductory texts [46, 47, 273,436], manuals [12, 14, 17, 315, 403, 421], and reviews [31, 42, 388] also are use-ful. The detailed survey of the HamiltonianKAM Theory presented in [17] isespecially recommended.

Finally it should be mentioned that we will not touch the so-called converseKAM Theory in this survey. The converse theorems assert that under appropriatehypotheses, dynamical systems admit no invariant quasi-periodic tori or the mea-sure of the union of those tori is small. The papers [200, 236,288, 290, 291, 298,394,414] exemplify this theory, see also [68,208,210,390].

2 Complex linearization

We deal with the linearization problem for a holomorphic mapnear a fixed point,for a description see V.I. Arnold’s manual [11] or J.W. Milnor’s monograph [304].

To be precise, consider a local holomorphic map (or a germ)F : (C, 0) →(C, 0) of the formF (z) = λz + f(z) with f(0) = f ′(0) = 0. The question iswhether there exists a local biholomorphic transformationΦ : (C, 0) → (C, 0)such that

Φ F = λ · Φ. (1)

We say thatΦ linearizesF near its fixed point0.

2.1 Formal solution

First consider the problem at a formal level. Given a series expansionf(z) =∑j>2 fjz

j we look for another seriesΦ(z) = z +∑

j>2 φjzj , such that the conju-

gation relation (1) holds formally. It turns out that a formal solution exists when-everλ 6= 0 is not a root of1 (clearly, it suffices to consider the case0 < |λ| 6 1,otherwise one can examineF−1 in place ofF ). Indeed, the coefficientsφj can be

in the case whereG is not an involution (i.e.,G2 is not the identity transformation). For generalreferences on reversible dynamical systems, see [251,356].

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determined recursively by the following equations:

λ(1 − λ)φ2 = f2,

λ(1 − λ2)φ3 = f3 + 2λf2φ2, (2)

λ(1 − λn−1)φn = fn + already known terms, n > 3.

From this the claim directly follows.

2.2 Convergence and small divisors

In the hyperbolic case where0 < |λ| 6= 1, the series forΦ has positive radius ofconvergence. This was proven by Poincare, not by considering the series but bya direct iteration method, compare with [11]. So there remains the elliptic casewith λ ∈ T1 ⊂ C, the unit circle on the complex plane. What is important foran analysis of equations (2) in this case is that, even ifλ is not a root of unity, itspowers do accumulate on1. This would givesmall divisorsin the formal series ofΦ, which casts doubt on its convergence.

This problem was successfully solved by C.L. Siegel [402] in1942. To this pur-pose, writingλ = e2πiβ, the followingDiophantine conditionswere introduced:for someτ > 1 andγ > 0 it is required that

∣∣∣∣β − p

q

∣∣∣∣ >γ

qτ+1, (3)

for all rationalsp/q (with q > 0). It turns out that this is sufficient for convergenceof the formal solution forΦ.

For the moment let it be enough to say that the set of allλ ∈ T1, such that thecorrespondingβ are Diophantine for someτ > 1 andγ > 0, has full measure inT1. In the next section we shall give a more elaborate discussion on Diophantinesets.

2.3 Measure and category

From J.C. Oxtoby’s manual [334] it is known that the real number lineR containssubsets that are large in measure and yet topologically small. The following ex-ample, due to H. Cremer [128] in 1927, illustrates this in thepresent situation. Fora nice description of this example in a somewhat different context see [26].

EXAMPLE 1 (Linearization of a quadratic map) [128]. Consider the mapF : C →C given by

F (z) = λz + z2,

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whereλ ∈ T1 is not a root of unity. We shall see that there is a topologicallylarge subset ofλ for which this map has periodic points in any neighborhood of0. For suchλ it follows that the formal conjugation diverges. Indeed, since thelinear mapz 7→ λz is the rotation over an angle incommensurable with2π andall the orbits (everywhere dense inT1) of such a rotation are infinite for|z| > 0,the existence of periodic orbits in every neighborhood of0 implies that the formallinearization must have zero radius of convergence.

To examine periodic points of periodq, we consider the equation

F q(z) = z. (4)

Using thatF q(z) = λqz + · · ·+ z2q

,

it follows thatF q(z) − z = z

(λq − 1 + · · ·+ z2q−1

).

AbbreviatingN(q) = 2q − 1, let z1, z2, . . . , zN be the non-zero solutions of (4).Their product satifies the relation

z1z2 · · · zN = 1 − λq,

sinceN(q) is odd. From this we see that there exists at least one solution withinradius

|λq − 1|1/N(q)

of z = 0. Now consider the set ofλ ∈ T1 satisfying

lim infq→∞

|λq − 1|1/N(q) = 0.

This set turns out to be residual,2 in fact, it resembles the set of Liouville numbers(we recall that a residual set contains a countable intersection of dense-open sets,which expresses that the set is large in the sense of topology). Notice that thisset is necessarily of measure zero, since it has all Diophantine numbers in itscomplement.

We conclude that for allλ contained in this residual set, periodic points ofFoccur in any neighborhood ofz = 0, which implies that for suchλ the formalnormal form transformation has zero radius of convergence.

2In other words: a denseGδ or a set of second Baire category [334].

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2.4 Discussion

By the end of the 20th century J.-C. Yoccoz [450, 452] completely solved theelliptic case, based on the so-calledBruno condition[89, 90, 364] on associatedcontinued fractions. To be precise, letpj/qj ∈ Q, j ∈ N, be the sequence ofconvergents3 to β /∈ Q. The Bruno condition then requires

∑

j>1

log qj+1

qj< +∞. (5)

For further information see e.g. Milnor’s monograph [304],also see Devaney’sarticle [141].

The above discussion reveals that in the setting of holomorphic maps, thereis nontrivial occurrence of dynamics that, up to a biholomorphic change of vari-ables, is a rigid rotation over a Diophantine irrational angle4 on an open regionof the complex plane (a neighborhood of0). In the above case this occurs on so-called Siegel discs. Similar results hold for so-called Arnold–Herman rings [204],compare with [156, 287]. For an overview of holomorphic dynamics, see [304].For other matters on holomorphic dynamics, compare with [141].

3 KAM Theory for circle and annulus maps

In the previous section we witnessed a typical occurrence ofrigid rotations onopen regions of the complex plane. Slightly paraphrasing this, we might say thatthe unperturbed system is a rigid rotation over a Diophantine angle, which corre-sponds to a large measure set in the parameter spaceT1 = λ. Siegel’s theo-rem [11,204,304,402] implies that this situation is persistent under all sufficientlysmall perturbations.

Keeping this in mind, we now turn to maps of the circle that areclose to rigidrotations in a suitable topology on the corresponding function space. For simplic-ity all future results are formulated in theC∞-topology [214, 318]. We mentionthat these topologies are compatible with the real analyticcase under the compact-open topology on holomorphic extensions [86]. Moreover, the present theory ad-mits generalizations in theCk-topology fork ∈ N sufficiently large. For details,see§ 4.2.1.

3The convergentsare the initial segments of a continued fraction. Ifβ = [a0; a1, a2, . . .] isthe continued fraction representation ofβ (in our casea0 = 0 since0 < β < 1) then thejthconvergent toβ is pj/qj = [a0; a1, a2, . . . , aj ]. The theory of continued fractions is expounded ine.g. [235,358].

4Strictly speaking, here and henceforth the words ‘rational’, ‘irrational’, ‘Diophantine’ refernot to the angle2πβ itself but to the numberβ.

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One important feature of the present problem is that parameters are neededfor persistence of conjugations to rigid rotations. Therefore we shall speak of1-parameter families of circle maps, that will be regarded as ‘vertical’ maps of thecylinder, i.e., as circle bundle maps.

3.1 Circle maps

Consider aC∞-family of circle diffeomorphisms

Pβ,ε : T1 → T1; x 7→ x + 2πβ + εa(x, β, ε), (6)

whereβ is a real parameter varying over an open finite intervalΓ ⊂ R. We regardε as a perturbation parameter, on which (6) also depends smoothly. Forε = 0 wedeal with the familyPβ,0 of rigid circle rotations.

For convenience we rewrite the family (6) as a ‘vertical’ cylinder map

Pε : T1 × Γ → T1 × Γ; (x, β) 7→(x + 2πβ + εa(x, β, ε), β

).

Naively speaking, the problem is to find a diffeomorphismΦε that conjugatesPε

andP0. To be precise, we require that the following diagram commutes:

T1 × ΓPε−−−→ T1 × ΓxΦε

xΦε

T1 × ΓP0−−−→ T1 × Γ,

meaning thatPε Φε = Φε P0, (7)

compare with (1).

3.1.1 Small divisors again

We proceed by formally solving equation (7). In order to respect the ‘verticality’of the cylinder mapsP0 andPε, we assume thatΦε has the skew form

Φε(x, β) =(x + εU(x, β, ε), β + εσ(β, ε)

), (8)

i.e., that it preserves projections to the parameter spaceΓ. It follows that (7) canbe rewritten as

U(x + 2πβ, β, ε)− U(x, β, ε)

= 2πσ(β, ε) + a(x + εU(x, β, ε), β + εσ(β, ε), ε

).

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Expanding in powers ofε and comparing lowest order coefficients, equation (7)leads to the linear equation

U0(x + 2πβ, β) − U0(x, β) = 2πσ0(β) + a0(x, β),

also called thehomologicalequation. The latter equation can be directly solvedin Fourier series. Indeed, writing

a0(x, β) =∑

k∈Z

a0k(β)eikx and

U0(x, β) =∑

k∈Z

U0k(β)eikx,

we find thatσ0 = − 12π

a00, which yields a parameter shift, and that, fork ∈ Z\0,

U0k(β) =a0k(β)

e2πikβ − 1,

while U00(β) can be taken arbitrarily, which roughly corresponds to a circle trans-lation. As in§ 2.2, we observe that generally there exists a formal solution onlyfor irrationalβ. As before, the powers ofe2πiβ still accumulate on1. This givessmall divisorsin the Fourier series, which threatens its convergence.

3.1.2 AKAM theorem for circle maps

To overcome the problem of small divisors,Diophantine conditionsare introducedas before, see (3), requiring that for given numbersτ > 1 andγ > 0, and for allrationalsp/q with q > 0 one has

∣∣∣∣β − p

q

∣∣∣∣ >γ

qτ+1. (9)

Let us denote the set of all suchβ by Rτ,γ ⊂ R, noting thatRτ,γ is closed.Recalling thatΓ ⊂ R is an open interval, we also consider the closed intervalΓγ =

β ∈ Γ | dist(β, ∂Γ) > γ

and nextΓτ,γ = Γγ ∩Rτ,γ. It follows thatΓτ,γ is

closed (and hence even compact). Thus, by the Cantor–Bendixson theorem [203]it follows thatΓτ,γ is the union of a perfect and a countable set. The perfect set is aCantor set, since it is also compact and totally disconnected (or zero-dimensional).The latter holds since the dense set of rationals is in the complement ofΓτ,γ. Weconclude thatΓτ,γ is nowhere dense and hence topologically ‘small’. However,the Lebesgue measure ofΓτ,γ, for γ ↓ 0, is large. Indeed, a small computationshows that

meas(Γ \ Γτ,γ

)6 const γ

∑

q>1

q−τ = O(γ), (10)

14

Φε

P0

Pε

x

x

β

β

Γτ,γ

Figure 1: Conjugation between the mapsP0 andPε onT1 × Γτ,γ.

asγ ↓ 0, where we have used the fact thatτ > 1, compare with e.g. [11, 67–69,223]. Note that the estimate (10) implies that the union

⋃

τ,γ

Γτ,γ

is of full measure inΓ, compare with remarks in the previous section. Again werefer to [334] for a discussion of measure vs category. As a first KAM theorem wenow formulate

THEOREM 2 (KAM for circle maps).In the above circumstances assume thatτ >1 and thatγ > 0 is sufficiently small. Then, if the familyPε is sufficiently closeto P0 in theC∞-topology, there exists aC∞-diffeomorphism of the cylinderΦε :T1 × Γ → T1 × Γ, of the skew form(8) with the following properties.

1. Φε is aC∞-near the identity map and dependsC∞-ly on ε.

15

2. The image of theP0-invariant union of circlesT1 × Γτ,γ underΦε is Pε-invariant, and the restricted mapΦε = Φε |T1×Γτ,γ

conjugatesP0 to Pε, thatis

Pε Φε = Φε P0.

ThatΦε is aC∞-near the identity map means that wheneverPε → P0 in theC∞-topology, alsoΦε → Id in theC∞-topology (here and henceforth, Id denotesthe identity mapping). Theorem 2 goes back to V.I. Arnold [3], also comparewith [11]. The present formulation closely fits with [67–69,223]. Concerningthe smoothness of the mapΦε, compare with J. Poschel’s results [346] and with[439,456,457].

The proof of Theorem 2 does not directly deal with the power series inε. In-stead it uses a Newton-like iterative method and an approximation property (byanalytic maps) of Whitney smooth maps defined on closed sets.Finally, the dif-feomorphismΦε is obtained by the Whitney Extension Theorem [438,439]. Thisapproach does need a formulation of the presentKAM theorem in terms of verticalcylinder maps, compare with Figure 1.

REMARKS.

1. For sufficiently smallγ > 0, the maximal size of the perturbation (the sizeof the difference betweenPε andP0) depends onγ in a linear way. Fromthe measure-theoretical point of view, it is optimal to chooseγ as smallas the perturbation allows. In certain cases it is even possible to considerthe limit asγ ↓ 0, thereby creating so-called Lebesgue density points ofquasi-periodicity [68,309,346].

2. Theorem 2, like many otherKAM theorems, has a perturbative character,since it only applies to small perturbations of the rigid rotation familyP0.In contrast to this, M.R. Herman [204,209] and J.-C. Yoccoz [448,450] haveproven a non-perturbative version of this theorem, in termsof the rotationnumber being Diophantine (also see [296]).

3. No KAM theorem in this survey (except for Theorem 19) will be providedwith a proof. The standard scheme of proving variousKAM statements, asinvented by Kolmogorov [238] and refined further by Arnold [6] and Moser[309] and subsequently by many other authors, is based on constructing aninfinite sequence of coordinate transformations whose domains of defini-tion shrink down to the invariant tori sought for. This cumbersome iterativeprocedure is similar to Newton’s method of tangents for solving algebraicequations. There are also approaches using ‘hard’ ImplicitFunction Theo-rems in infinite-dimensional spaces (see e.g. [457]) or the Schauder fixed-point theory (see e.g. [208]). Recently, the so-called ‘direct methods’ in

16

proving the existence and persistence theorems for quasi-periodic motionswere developed that deal directly with (Poincare–Lindstedt) series in theperturbation parameter and exploit techniques usual in Quantum Field The-ory, like the multiscale decomposition, tree expansions, and renormalizationgroups (see e.g. [30,37,110–112,153,161,169,170,173–175,177–179] andreferences therein). These techniques allow one to find explicitly delicatecancellations (‘compensations’) among large terms of the Lindstedt series(absolutely divergent due to the small divisors) and obtainestimates imply-ing convergence.

3.1.3 Discussion

A circle diffeomorphism smoothly conjugated to a rigid rotation Pβ,0 : x 7→ x +2πβ with β irrational is said to bequasi-periodic. It is well-known that each orbitof such a map fills the circle densely, see e.g. [11, 12, 140]. For β ∈ Γτ,γ, therigid rotationPβ,0 certainly is quasi-periodic. A first consequence of Theorem2is that the circle mapsPβ,ε that are conjugated to one of the Diophantine rigidrotationsPβ′,0, are still quasi-periodic. In fact, sinceΦε is near identity in theC∞-topology, it follows that, forε 6= 0 small, the measure of the union of circles withDiophantine rotation numbers is still large (for a definition of the rotation numbersee e.g. [11, 140]). The conclusion is that quasi-periodicity typically occurs withpositive measure in the parameter space. Moreover, the factthat a Cantor set isperfect, meaning that it contains no isolated points, implies that quasi-periodicityalmost neveroccurs as an isolated phenomenon.

We formulated Theorem 2 in its (structural) stability form,which for this oc-casion is calledquasi-periodic stability[67–69, 223]. This term was chosen inreminiscence of the so-calledΩ-stability, which refers to structural stability whenrestricted to the non-wandering set [234].

The Arnold family. An example is given by the Arnold family

Pβ,ε(x) = x + 2πβ + ε sinx (11)

of circle maps, where we consider the(β, ε)-plane of parameters. We restrict to|ε| < 1, to ensure thatPε is a diffeomorphism. For|ε| 1 the familyPε is closeto P0 in theC∞-topology. This even holds in the compact-open topology on holo-morphic extensions, as mentioned earlier. It is known [3,11,53,79,140] that fromthe points(β, ε) = (p/q, 0), resonance tongues (otherwise calledArnold tongues)emanate into the two open half-planesε 6= 0, in such a way that for smallε anopen and dense subset is covered. In the tongue emanating from (β, ε) = (p/q, 0),the dynamics is asymptotically periodic with rotation numberp/q. Compare withFigure 2.

17

3

2.5

2

1.5

1

0.5

1/100 1/9 1/8 1/7 1/6 1/5 2/9 1/4 2/7 3/10 1/3 3/8 2/5 3/7 4/9 1 2/

0

ε

β

Figure 2: Resonance tongues in the Arnold family [79].

Theorem 2 implies that in the complement of this union of tongues, there existsa union of smooth curves that fill out positive measure. For parameter values onthese curves, the dynamics is Diophantine quasi-periodic.

Applications. Theorem 2 has applications for systems of ordinary differentialequations with a2-torus attractor, of whichPε is a Poincare map close to a rigidrotation family. In that case the Diophantine quasi-periodic subsystem ofPε isreferred to as a family ofquasi-periodic attractors[361].

The basic ingredient of the examples to follow is a nonlinearoscillator withequation of motion

u + cu + au + f(u, u) = 0, (12)

whereu ∈ R andu = du/dt, which is assumed to have a hyperbolic periodic at-tractor, i.e., a periodic solution with a negative Floquet exponent. For the momentwe consider coefficients likea andc as positive constants, but later on some ofthem occasionally will act as parameters. A classical example of such a systemis the Van der Pol oscillator, where the nonlinearity is given by f(u, u) = bu2u,with b a real constant.

As a first example with quasi-periodic attractors, considerthe oscillator (12)subject to a weak time-periodic forcing:

u + cu + au + f(u, u) = εg(u, u, t), (13)

18

u, t ∈ R, whereg(u, u, t+2π/Ω) ≡ g(u, u, t), and whereε is a small perturbationparameter. As usual we take the timet as an extra state variable, introducing the3-dimensional (generalized) phase spaceR2 × T1 with coordinates(u, u) ∈ R2

andt ∈ T1 = R/(

(2π/Ω)Z). Here the non-autonomous oscillator (13) defines

the vector fieldXε given by

u = v

v = −au − cv − f(u, v) + εg(u, v, t) (14)

t = 1.

In the unperturbed caseε = 0 the oscillator is free and in (14) decouples fromthe third equationt = 1. Combining the periodic attractor of the free oscilla-tor with this third equation gives rise to an invariant2-torus attractor to be de-noted byT0. By direct techniques from ordinary differential equations [11, 191]one can show that the Poincare return map or stroboscopic map of the sectiont = 0 mod (2π/Ω)Z has the appropriate circle map format forCk-versions ofTheorem 2 (fork ∈ N large), for details compare with, e.g., [52, 68]. Herewe use the fact that the torus attractorT0 is normally hyperbolic by hyperbol-icity of the periodic orbit. Thus, according to the Center Manifold Theorem,see [117,162,215,424],T0 persists as an invariant manifold. This means that, for|ε| 1, the vector fieldXε has a smooth invariant2-torusTε (close toT0), alsodepending smoothly onε. Here ‘smooth’ means ‘Ck’ for an appropriatek ∈ N,which in this case tends to∞ asε → 0. Note that we need to regard a coeffi-cient likea in (13) as a parameter, to obtain a family of quasi-periodic attractorsin the Poincare map. We summarize by saying that Theorem 2 provides a familyof quasi-periodic attractors in the system (14).

As a second example consider two nonlinear oscillators witha weak coupling

u1 + c1u1 + a1u1 + f1(u1, u1) = εg1(u1, u2, u1, u2)

u2 + c2u2 + a2u2 + f2(u2, u2) = εg2(u1, u2, u1, u2),

u1, u2 ∈ R. This yields the following vector fieldXε on the4-dimensional phasespaceR2 × R2 =

(u1, v1; u2, v2)

:

uj = vj

vj = −ajuj − cjvj − fj(uj, vj) + εgj(u1, u2, v1, v2),

j = 1, 2. Note that forε = 0 the system decouples to a system of two independentoscillators, which has an attractor in the form of a2-torusT0. This torus arisesas the product of two circles, along which each of the oscillators has its periodicsolution. (The circles lie in the two-dimensional planes given byv2 = u2 = 0

19

andv1 = u1 = 0 respectively.) The persistence ofT0 for |ε| 1 runs exactlylike before and the Poincare map is defined accordingly, again yielding a familyof quasi-periodic attractors.

It may be clear that a similar coupling ofn nonlinear oscillators gives rise toan attractingn-torus insideR2n. Below we shall obtain a formulation for thisproblem (see Theorem 3), which is more appropriate for generalization to higherdimensions (compare with [14]). There are higher dimensional analogues of thepresent situation, where next to periodicity and quasi-periodicity, also chaotic dy-namics coexists [52, 68, 69]. This scenario and the transitions or bifurcations be-tween the various kinds of dynamics have been associated with the onset of turbu-lence in fluid dynamics, see Ruelle, Takens, and Newhouse [326, 362, 363]. Thequasi-periodic state then is seen as intermediate between very orderly and chaotic,also see [217,252,253]. In§ 4.4.2 we shall return to this subject.

3.2 Area preserving annulus maps

Close to the setting of circle maps (or of ‘vertical’ cylinder maps), is that of mapsof the annulus that preserve area. This set-up relates to conservative dynamics bythe Liouville Theorem [12,17]. First we develop Moser’s Twist Mapping Theorem[309,316] as an analogue of Theorem 2.

3.2.1 Moser’s Twist Mapping Theorem

Consider an annulus with ‘polar’ coordinates(x, y) ∈ T1 × A, whereA ⊂ R isopen. We endowT1 × A with the area formσ = dx ∧ dy. Consider aC∞-mapPε : T1 × A → T1 × A of the form

Pε(x, y) =(x + 2πβ(y), y

)+ ε(f(x, y), g(x, y)

), (15)

that preserves the areaσ, meaning thatdet DPε ≡ 1. Note that forε = 0 thismap leaves the family of circlesy = const invariant, and again the problem is thepersistence of this family whenPε is C∞-nearP0, i.e., when|ε| 1. We saythatP0 is a (pure) twist map if the mapy 7→ β(y) is strictly monotonic5 (hence adiffeomorphism) onA. We impose Diophantine conditions as before, see (3) and(9). To be more precise, for given constantsτ > 1 andγ > 0 we require againthat ∣∣∣∣β(y) − p

q

∣∣∣∣ >γ

qτ+1, (16)

for all rationalsp/q with q > 0, or, in other words, thatβ(y) ∈ Rτ,γ. DefineΓ = β(A) as well as subsetsΓγ andΓτ,γ ⊂ Γ, as in§ 3.1.2. We note that the map

5Or, equivalently, ifdβ/dy is nowhere zero.

20

β pullsΓτ,γ back to a subsetAτ,γ ⊂ A, which, forγ > 0 sufficiently small, ‘is’ aCantor set of large measure.

THEOREM 3 (Twist). In the above circumstances assume thatτ > 1 and thatγ > 0 is sufficiently small. Then, ifPε is sufficiently close toP0 in the C∞-topology, there exists aC∞-diffeomorphism of the annulusΦε : T1×A → T1×Awith the following properties.

1. Φε is aC∞-near the identity map and dependsC∞-ly on ε.

2. The image of theP0-invariant union of circlesT1 × Aτ,γ underΦε is Pε-invariant, and the restricted mapΦε = Φε |T1×Aτ,γ

conjugatesP0 to Pε,that is

Pε Φε = Φε P0.

This theorem was first proven by J.K. Moser [309] for maps of the classC333,for additional comments see [316]. Subsequently, H. Russmann reducedC333 toC5 [365], while F. Takens proved thatC1 is not enough [414]. As was finallyshown by M.R. Herman [208], Theorem 3 is carried over toC3-mappings butnot to C2-mappings whose second derivatives belong to the Holder classC1−δ,however smallδ > 0 is. The present formulation is close to Theorem 2 and,concerning the smoothness ofΦε, the same comments apply again. For a niceexposition of the real analytic version of Theorem 3 (accompanied by a completeproof), see [403].

REMARK . The remark on the perturbation size depending onγ (see the firstremark after the formulation of Theorem 2) also applies here. One example wherethe limit asγ ↓ 0 can be taken, is near a generic elliptic fixed point, which therebybecomes a density point of quasi-periodicity. This situation often is referred to as‘small twist’ [309,346].

3.2.2 Discussion

Theorem 3 (on the annulus) is quite close to Theorem 2 (on the cylinder), wherethe role of the parameterβ has been taken by the action variabley. In the samespirit as in§ 3.1.3, we conclude that for the area preserving case, typically quasi-periodicity occurs with positive measure in the phase space.

A difference in the settings of Theorems 2 and 3 is that the former deals with‘vertical’ maps while the latter does not. This means that generally in Theorem 3projections to the action spaceA = y arenot preserved by the conjugationΦε.Moreover,Φε generally is not symplectic.

21

Applications. We mimic the dissipative discussion presented in§ 3.1.3. As afirst example consider the mathematical (planar) pendulum with a (weak) periodicforcing. As possible equations of motion one may take

u + ω2 sin u = ε cos t or

u + (ω2 + ε cos t) sin u = 0,

which give rise to volume preserving3-dimensional vector fields. For simplicitywe only write down the first example:

u = v

v = −ω2 sin u + ε cos t

t = 1.

As is usual in mechanics, see e.g. [12, 14, 17] as well as [129,297], we introduceangle-action variables(x, y) for ε = 0, i.e., for the autonomous planar pendu-lum.6 In fact, denoting the energy byH0(u, v) = 1

2v2 − ω2 cos u, we restrict

ourselves to the oscillatory region whereH0(u, v) < ω2. Next consider any levelsetH0(u, v) = h

, with |h| < ω2. The action variabley then is defined by

y(h) =1

2π

∮

H0(u,v)=h

v du,

which is proportional to the area enclosed by the level set. The angle variablexis obtained by taking the time parametrization of the periodic motion inside thislevel set scaled to period2π. Thus one obtains the canonical equations

y = 0, x = β(y)

for the oscillatory motions of the planar pendulum.One easily sees that the Poincare (or stroboscopic) mapPε for a perturbed pen-

dulum has the form (15) and is area preserving. A direct computation, involvingan elliptic integral, shows thatP0 is a pure twist map. According to Theorem 3,the conclusion of quasi-periodicity occurring with positive measure in the phasespace does apply here.

A related application deals with two coupled oscillators

u1 = −ω21 sin u1 − ε

∂U

∂u1(u1, u2)

u2 = −ω22 sin u2 − ε

∂U

∂u2(u1, u2),

6In the literature, such angle-action variables often are denoted by(x, y) = (ϕ, I) or (α, a).

22

leading to a4-dimensional Hamiltonian vector field

uj = vj

vj = −ω2j sin uj − ε

∂U

∂uj(u1, u2),

j = 1, 2, with the Hamilton function

Hε(u1, u2, v1, v2) = 12v21 + 1

2v22 − ω2

1 cos u1 − ω22 cos u2 + εU(u1, u2).

In this case, it is the iso-energetic Poincare maps that obtain the formPε of (15).Now Theorem 3 yields the conclusion of quasi-periodicity occurring with positivemeasure in the energy hypersurfaces ofHε.

This discussion also leads to higher dimensional explorations inKAM Theory.

4 KAM Theory for flows

We turn to the context of smooth vector fields on manifolds (locally correspondingto systems of ordinary differential equations). First we give a formal definition ofquasi-periodicity, next considering a few conceptual aspects.

4.1 Introduction

Let M be aC∞-manifold andX aC∞-vector field onM . Fix n ∈ N with n > 2.Also consider the standardn-torusTn = Rn/(2πZ)n endowed with coordinatesx1, x2, . . . , xn counted modulo2πZ. For any vectorω ∈ Rn, onTn we considerthe constant vector field

Xω =n∑

j=1

ωj∂

∂xj, (17)

in the system form given by

xj = ωj, 1 6 j 6 n.

Assume thatT ⊆ M is anX-invariantn-torus. We say that the restrictionX|Tis parallel or conditionally periodicwhenever there exist a vectorω ∈ Rn and aC∞-diffeomorphismΦ : T → Tn conjugatingX|T with a constant vector fieldXω, which means thatΦ∗(X|T ) = Xω. We refer to theωj as thefrequenciesofX|T and toω = (ω1, ω2, . . . , ωn) as thefrequency vectorof X|T .

We first note thatω is not uniquely determined byT , but depends also onΦ. Ifwe composeΦ by a translation onTn, ω does not change. However, let us consideran invertible linear mapS : Rn → Rn, thenS projects to a torus diffeomorphism

23

if and only if S ∈ GL(n, Z).7 Now if Φ is composed byS, the frequency vectorchanges toSω, sinceS∗Xω = XSω.

The dynamics onX|T is said to bequasi-periodic(or non-resonant) wheneverthe frequenciesω1, ω2, . . . , ωn are independent over the rationalsQ, compare with§ 3.1.3. We also callT a quasi-periodic invariantn-torus ofX. Observe thatsuch a quasi-periodicn-torusT is densely filled by each of itsX-trajectories. Onthe other hand, if the frequenciesω1, ω2, . . . , ωn satisfyl independent resonancerelations (1 6 l 6 n), then the torusT is foliated into invariant(n − l)-tori of X(which are quasi-periodic forl 6 n− 2), and eachX-trajectory onT densely fillsone of these tori.

REMARKS.

1. Note that the definitions of parallel and quasi-periodic dynamics onT donot depend on the choice of the conjugationΦ. Moreover, the so-calledfrequency module(or lattice of frequencies)

L(ω) =ω1k1 + ω2k2 + · · ·+ ωnkn | k ∈ Zn

of an invariantn-torus with parallel dynamics and frequenciesω1, ω2, . . . , ωn

is determined uniquely [233, 311, 312]. To be more precise, it is not hardto verify thatL(ω) = L(ω′) if and only if there exists an operatorS ∈GL(n, Z) such thatω′ = Sω.

2. A similar formal definition of quasi-periodicity can be given for diffeo-morphisms of a torus. In the previous sections we implicitlyused such adefinition in the case of circle diffeomorphisms and of holomorphic maps,compare with§ 3.1.3.

3. The structure of orbits ofGL(n, Z) or of SL(n, Z) =S ∈ GL(n, Z) |

det S = 1

in Rn is known: J.S. Dani’s theorem [134] states that ifω ∈ Rn

is not proportional to an integer vector (i.e., if the trajectories ofXω are notclosed), then the orbitSL(n, Z)ω is densein Rn, and vice versa. Thereare generalizations of this result to the action of the groupSL(n, Z) on thespace ofd-frames inRn, 2 6 d 6 n − 1 [135, 186]. So, the set of allthe frequency vectors that can be assigned to a given conditionally periodicmotion on ann-torusT is in factdensein Rn, provided that this motion isnot ‘maximally’ resonant (see also a discussion in [400]).

7GL(n, Z) is the group of linear operators inRn represented by matrices with integer entriesand determinant±1.

24

ω1

ω2

Figure 3: Sketch of the setDτ,γ(R2).

4.1.1 Affine structure

We shall argue that on any quasi-periodicX-invariantn-torusT , a natural affinestructure is defined. To this end first observe that the self-conjugations of theconstant vector fieldXω, with rationally independent frequenciesω1, ω2, . . . , ωn,are exactly the translations of the standard torusTn. This directly follows fromthe fact that each trajectory ofXω is dense. Note that these translations determinethe affine structure onTn.

LEMMA 4 (Affine structure) [51,83].In the above setting with aC∞-diffeomorphismΦ : T → Tn such thatΦ∗(X|T ) = Xω, the self-conjugations ofX|T determine anatural affine structure onT . The translations ofT are self-conjugations and theconjugationΦ is unique modulo torus translations onT andTn.

REMARK . Note that the structure onT is a bit stronger than affine, since the tran-sition maps are not general elements ofGL(n, R), but are restricted toGL(n, Z).

4.1.2 The perturbation problem

We start preparing the perturbation problem. As before, forsimplicity we formu-late all the results in theC∞-topology [214,318].

25

We consider an unperturbed vector field asintegrable, when it is invariant un-der a suitable action of then-torus Tn (n > 2), whereas the invariant tori inquestion are orbits of this action, compare with [67–69, 223]. In this case we canrestrict our attention to a subset of the phase spaceM diffeomorphic toTn × Rm

(m > 0). We note that any (isolated) paralleln-torusT by definition is integrable.Now for anyA ⊂ Rm open (and bounded), we may consider theCk-norm

‖ · ‖k,A for C∞-functions on the closureTn ×A. TheC∞-topology onC∞(Tn ×Rm) then is generated by all such norms. This induces similar topologies on allspaces of smooth dynamical systems [214,318].

As in the previous section (see§§ 3.1.2 and 3.2.1), theKAM theorems will beformulated in terms of conjugations between subsystems of an integrable systemand its perturbation, which are nearby in theC∞-topology. This is the structuralstability formulation, for this occasion calledquasi-periodic stability, as alreadymentioned in§ 3.1.3. The subsystems will be (unions of) quasi-periodic invariantn-tori, so with non-resonant frequenciesω1, ω2, . . . , ωn. In fact and as before, seeinequalities (3), (9), (16), we shall need stronger nonresonance conditions to beintroduced now.

REMARK . In many applications, the open setA ⊂ Rm is given by a localnondegeneracy condition via the Inverse Function Theorem [214,318,407].

Diophantine conditions. The strong nonresonance conditions on the frequen-cies, as mentioned above, are Diophantine in the following sense. Letτ > n − 1andγ > 0 be constants. Set

Dτ,γ(Rn) =

ω ∈ Rn

∣∣ |〈ω, k〉| > γ|k|−τ , for all k ∈ Zn \ 0. (18)

Here and henceforth,

〈ω, k〉 =n∑

j=1

ωjkj and |k| =n∑

j=1

|kj|.

Elements ofDτ,γ(Rn) are called(τ, γ)-Diophantinefrequency vectors. One easily

sees thatDτ,γ(Rn) is a closed subset with the following closed half line property:

wheneverω ∈ Dτ,γ(Rn) ands > 1, then also the productsω ∈ Dτ,γ(R

n), comparewith Figure 3. The intersectionSτ,γ = Dτ,γ(R

n)∩ Sn−1 of Dτ,γ(Rn) with the unit

sphere again is a closed (even a compact) set. An applicationof the Cantor–Bendixson theorem [203] yields thatSτ,γ is the union of a perfect and a countableset. Since the resonant hyperplanes (with equations〈ω, k〉 = 0, k ∈ Zn \ 0)give a dense web in the complementSn−1 \ Sτ,γ, it follows that this perfect set istotally disconnected. Summing up we conclude that the perfect subset ofSτ,γ is a

26

Cantor set. This implies thatDτ,γ(Rn) is nowhere dense. Moreover, the measure

of Sn−1 \ Sτ,γ is of the order ofγ asγ ↓ 0, compare with the discussion on theDiophantine condition (9) in§ 3.1.2.

In Figure 3 we roughly sketch the setDτ,γ(R2). Considering the intersection

with the lineω2 = 1 directly gives the connection of (18) with (3), (9), and (16),compare with [123].

In the sequel, we shall regardτ as afixednumber greater thann − 1.

Mild Diophantine conditions. Diophantine conditions like (3), (9), (16), and(18), as well as future versions (30) (in§ 5.2.1), (33) (in§ 6.1.1), (36) (in§ 6.1.3),and (52) (in§ 8.3.1), serve to overcome the small divisor problems that are inher-ent toKAM Theory as described in§§ 2.2 and 3.1.1. For a more detailed treatmentof small divisors we refer ahead to§ 5.1.2.

In the literature (see e.g. [177, 347, 368–372]), also ‘milder’ Diophantine con-ditions occur, like

|〈ω, k〉| >γ

∆(|k|) for all k ∈ Zn \ 0.

Here∆ is anapproximation function, i.e., an arbitrary monotonically increasing(or just nondecreasing) continuous function∆ : [1,∞) → R+ = [0,∞) such that∆(1) > 0 and ∫ ∞

1

log ∆(u) du

u2< +∞.

Note that the usual Diophantine condition (18) correspondsto the case∆(u) = uτ .Yet another mild version of the Diophantine condition is given by [177]

∑

j>0

2−j log1

min0<|k|62j

|〈ω, k〉| < +∞,

which in reminiscence of (5) also is called the Bruno condition, compare with,e.g., [89,90,347,364,367–372,450,452].

For a similar discussion also see [68],§ 1.5.2.

Shrinking the frequency domain. Let Γ ⊂ Rn be an open subset of frequencyvectors. We also consider the shrunken version ofΓ defined by

Γγ =ω ∈ Γ | dist(ω, ∂Γ) > γ

.

Let Dτ,γ(Γγ) = Γγ ∩ Dτ,γ(Rn). In the sequel we always takeγ sufficiently small

to ensure that the nowhere dense setDτ,γ(Γγ) has positive measure. Recall thatthe measure ofΓ \Dτ,γ(Γγ) tends to zero inΓ asγ ↓ 0, compare with [6,7,11,68,69,223,346].

27

4.2 Families of normally hyperbolic quasi-periodic tori

Now we are in a position to deal with theKAM theorem for normally hyperbolicinvariant tori. By the Center Manifold Theory [117, 162, 215, 424] such tori,asinvariant manifolds, persist under small perturbations. We shall restrict to such‘center manifold’ tori, setting up the above perturbation program for this case. Werecall that parallelity of such a torus implies integrability.

So, we assume parallelity of certain unperturbed ‘center manifold’ tori, whichimplies∞-normal hyperbolicity. The perturbed tori then are diffeomorphic to theunperturbed ones. The diffeomorphism has a finite degree, say k, of smoothness[411]. Moreover, the perturbed tori are unique, i.e., independent ofk, and we cantakek → ∞ as the perturbation becomes small.

Summarizing, we assume that the phase space of both the unperturbed and theperturbed systems consists of then-torusTn, where, as usual, we taken > 2. Theinterest again is the persistence of parallelity. Referring to similar discussions in§ 3.1.3, we can see that we have to restrict to Diophantine quasi-periodic invarianttori. As said before at the beginning of Section 3, theKAM Theory in question hasCk-versions for sufficiently largek ∈ N, but for simplicity we formulate allKAM

theorems in the sequel in theC∞-setting, similarly to Theorems 2 and 3 above.As in the case of circle maps, see§ 3.1, we need parameters for persistence ofquasi-periodicn-tori.

4.2.1 Formulation of the normally hyperbolic KAM theorem

Let P ⊆ Rs be an open set of parameters and consider families of vector fieldsX = Xµ(x), with x ∈ Tn = Rn/(2πZ)n andµ ∈ P . We shall treat such a familyas a ‘vertical’ vector field on the productTn × P . Throughout we assume aC∞-dependence of all the vector fields on bothx andµ. As before we often use thevector field notation, writing

f(x, µ)∂

∂xinstead of x = f(x, µ),

compare with (17). The starting point is an integrable family

Xµ(x) = ω(µ)∂

∂x, (19)

x ∈ Tn, µ ∈ P , where integrability amounts tox-independence, which expressesinvariance under the naturalTn-action. Our interest is the family ofX-invariantn-tori Tn × µ, whereµ ∈ P . The smooth mapω : P → Rn is called thefrequency map.

The familyX is said to benondegenerateat µ0 ∈ P if the derivativeDµ0ω issurjective (in particular, this impliess > n). As in Section 3, we are especially

28

interested in the fate of theX-invariant toriTn × µ, µ ∈ P , under smoothperturbations

Xµ = Xµ + f(x, µ)∂

∂x, (20)

whenµ is nearµ0 and where the size ofX − X (i.e., the size off ) is small in theC∞-topology. Here we confine our attention to the tori that are Diophantine in thesense of (18).

By the Inverse Function Theorem [214,318,407], the nondegeneracy conditionimplies that the pointµ0 ∈ P has an open bounded neighborhoodA ⊆ P , re-stricted to which the mapω is a submersion, i.e., is conjugated to the projectionon a lower dimensional subspace. We then say thatX is nondegenerateonTn×A.DefiningΓ = ω(A), we consider shrunken versionsΓγ andAγ = ω−1(Γγ) of thedomainsΓ andA, respectively, see§ 4.1.2 above. Accordingly we consider theset of Diophantine frequency vectorsDτ,γ(Γγ) and its pull-backDτ,γ(Aγ) alongthe frequency map. Note that the measure ofA \ Dτ,γ(Aγ) tends to0 asγ ↓ 0.

THEOREM 5 (Normally hyperbolicKAM ) [67–69,123,223].Let n > 2. Let theintegrableC∞-family X = Xµ(x) of vector fields(19) be nondegenerate onTn × A, with A ⊆ P open. Then, forγ > 0 sufficiently small, there existsa neighborhoodO of X in theC∞-topology such that for any perturbed familyX ∈ O as in(20), there exists a mappingΦ : Tn×A → Tn×A with the followingproperties.

1. Φ is aC∞-diffeomorphism onto its image and aC∞-near the identity map.Also,Φ preserves projections toP .

2. The image of theX-invariant torus unionTn × Dτ,γ(Aγ) underΦ is X-invariant, and the restricted mapΦ = Φ |Tn×Dτ,γ(Aγ) conjugatesX to X,that is

Φ∗X = X.

REMARKS.

1. As in the case of Theorems 2 and 3, the fact thatΦ is aC∞-near the identitymap means that in theC∞-topology, wheneverX → X alsoΦ → Id.

2. The mapΦ takesTn×Aγ into Tn×A. Theorem 5 asserts that the integrablesystem (19) is (locally) quasi-periodically stable [67–69,223], compare withthe discussion in§ 3.1.3 following Theorem 2.

3. As in § 3.1, the diffeomorphismΦ is taken of a skew form, which meansthat the projectionTn × A → A is preserved, or in other words, that the

29

transformation in theµ-direction is independent of the anglesx ∈ Tn. Thistransformation in theµ-direction gives a diffeomorphic image of the unionof closed half lines inDτ,γ(Aγ).

4. There are direct generalizations of Theorem 5 to the worldof Ck-systemsendowed with theCk-topology [214, 318] fork ∈ N sufficiently large. ForCk-versions of the presentKAM Theory, see [109, 115, 208, 256, 257, 309,346, 365, 374, 375, 413, 457], for discussions on this subject also see [7, 68,69, 223, 316, 397]. To give an idea, fork > 4τ + 2, the conjugation isat least of classCk−2τ , also compare with§ 5.1.2 below. Therefore in theC∞-case, no losses of differentiability occur and the conjugations also areof classC∞. In the real analytic case, the conjugations are even Gevreysmooth [342,343,399,431,446,459]. More generally, the conjugations alsoare Gevrey smooth as soon as the original system is [344,345,460].

4.2.2 Discussion

In Theorem 5, the restriction ofΦ to the union of the Diophantine quasi-periodictori Tn ×Dτ,γ(Aγ) preserves the natural affine structure of the quasi-periodic tori,see§ 4.1.1. In the complement of this nowhere dense set the diffeomorphismΦhasno dynamical meaning.

The set-up (and proof) of Theorem 5 is very close to the ‘classical’ KAM the-orem for Lagrangean invariant tori in nearly integrable Hamiltonian systems (see§ 4.3) in the formulation of J. Poschel [346], also compare with [67–69,123,223].In fact Theorem 5 relates to this ‘classical’KAM theorem as Theorem 2 does toTheorem 3.

The normally hyperbolicKAM Theorem 5 particularly is relevant in the casewhere the tori are attractors, in which case we are dealing with families of quasi-periodic attractors[361]. In the discussion in§ 3.1.3 following Theorem 2, weconsidered the casen = 2, where generically on the tori in between the quasi-periodic ones, only regular dynamics could occur with a Poincare map that isKupka–Smale [335]. In particular this refers to the oscillator models in§ 3.1.3.Forn > 3 the situation ‘in between’ the quasi-periodic tori can be a little different,since invariant3-tori can contain strange attractors [326].

Below in Section 6 we shall discuss several scenarios where families of quasi-periodic attractors undergo bifurcations. One of these is the quasi-periodic Hopfbifurcation, where the torus attractors lose stability anda higher dimensional torusfamily branches off.

Computational and numerical aspects of normally hyperbolic quasi-periodictori have been dealt with in [79,93–95,379], also see [405].

30

4.3 KAM Theory for Lagrangean tori in Hamiltonian systems

We switch to the world of Hamiltonian systems of classC∞. This means, first ofall, that the phase spaceM is now a symplectic manifold, say of dimension2n.First consider a Liouville integrable Hamiltonian system with n degrees of free-dom, recalling that this means that there existn independent integrals in involu-tion, including the energyH. If the energy level locally is compact, the Liouville–Arnold Integrability Theorem [5,12,14,17,129,297] provides us with angle-actionvariables(x, y) ∈ Tn ×Rn. This implies that the corresponding Hamiltonian vec-tor field X = XH possessesn-parameter families of Lagrangean invariantn-toriTy, parametrized byy. The word ‘Lagrangean’ means that the dimension of thetori is equal to the number of degrees of freedom and that the restriction of thesymplectic form to these tori vanishes.

This situation fits well in the general approach of this section, since the Liouville–Arnold Integrability Theorem and its proof provide a smoothconjugation ofXH |Ty

and a constant vector field on the standard torusTn, as well as an affine structurethat fits with Lemma 4. Note that in this Hamiltonian setting,both the conjugationand the affine structure extend to all parallel (or conditionally periodic) invariantn-tori.

4.3.1 Formulation of the LagrangeanKAM theorem

Forn > 2 considerP ⊆ Rn as an open domain. As before, letTn = Rn/(2πZ)n

denote the standardn-torus. The productM = Tn × P is endowed with coordi-nates(x, y) = (x1, x2, . . . , xn, y1, y2, . . . , yn), where thexj are counted modulo2πZ, and with the symplectic formσ = dx ∧ dy =

∑nj=1 dxj ∧ dyj .

Assume that the HamiltonianH : M → R does not depend on the angle vari-ablex. The corresponding Hamiltonian vector fieldXH , defined by the relationιXH

σ = dH,8 then takes the form

XH(x, y) = ω(y)∂

∂x=

n∑

j=1

ωj(y)∂

∂xj, (21)

whereω(y) = ∂H(y)/∂y is the frequency vector. Formula (21) expresses that(x, y) is a collection of angle-action variables forXH [2, 12, 14, 17, 84, 129]. Asin § 4.2.1, we callω : P → Rn the frequency map, saying thatH is Kolmogorovnondegenerateaty0 ∈ P if the derivativeDy0ω is invertible.

As in §§ 3.1, 3.2, 4.2, we are particularly interested in the fate of theX-invariantLagrangean toriTn × y, y ∈ P , under smooth Hamiltonian perturbationsX of

8Recall that(ιXσ)(Y ) = σ(X, Y ) for any vector fieldY [2,12,17,68,84,352]. In the literature,Hamiltonian vector fieldsXH are often defined byιXH

σ = −dH .

31

X = XH , wheny is neary0 and where the size ofX − X is small in theC∞-topology. Again as before, we confine our attention to the tori that are Diophantinein the sense of (18).

By the Implicit Function Theorem [214,318,407], the nondegeneracy conditionimplies thaty0 ∈ P has an open bounded neighborhoodA ⊆ P , restricted towhich the mapω is a diffeomorphism. We then say thatX = XH is Kolmogorovnondegenerateon Tn × A. DefiningΓ = ω(A), we consider shrunken versionsΓγ andAγ = ω−1(Γγ) of the domainsΓ andA, respectively, see§ 4.1.2 above.Accordingly we consider the set of Diophantine frequency vectorsDτ,γ(Γγ) andits pull-backDτ,γ(Aγ) along the frequency map. Note that the frequency mappulls back the union of closed half lines inDτ,γ(Γγ) in a diffeomorphic way. Alsonote that the measure ofA \ Dτ,γ(Aγ) tends to0 asγ ↓ 0.

Staying within the class of Hamiltonian systems, we perturbthe Hamiltonianvector fieldX = XH in theC∞-topology as described in§ 4.1.2.

THEOREM 6 (LagrangeanKAM ) [67–69,223,346].Suppose that the integrable Hamil-tonianC∞-systemX = XH is Kolmogorov nondegenerate onTn ×A, for A ⊆ Popen. Then, for sufficiently smallγ > 0, there exists a neighborhoodO of X intheC∞-topology such that for each Hamiltonian vector fieldX ∈ O, there existsa mappingΦ : Tn × A → Tn × A with the following properties.

1. Φ is aC∞-diffeomorphism onto its image and aC∞-near the identity map.

2. The image of theX-invariant torus unionTn × Dτ,γ(Aγ) underΦ is X-invariant, and the restricted mapΦ = Φ |Tn×Dτ,γ(Aγ) conjugatesX to X,that is

Φ∗X = X.

REMARKS.

1. The mapΦ, that takesTn×Aγ intoTn×A, generally is not symplectic. The-orem 6 asserts that the integrable system (21) is (locally) quasi-periodicallystable [67–69,223], compare with the discussion in§ 3.1.3 following Theo-rem 2 and with Remark 2 in§ 4.2.1 following Theorem 5.

2. Unlike in the case of Theorem 5, here the projectionTn × A → A is notpreserved by the mapΦ. Nevertheless, the perturbed union of invariant toriΦ(Tn ×Dτ,γ(Aγ)

), up to a diffeomorphism, is organized in terms of closed

half lines as described in Remark 3 following Theorem 5.

3. The discussion onCk-generalizations of Theorem 6, fork ∈ N sufficientlylarge, runs as in Remark 4 following Theorem 5.

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4.3.2 Discussion

We now discuss several variations on the ‘classical’ LagrangeanKAM Theorem 6.

A typicality conclusion. Theorem 6 is the ‘classical’KAM theorem in its stabil-ity formulation, for earlier formulations see e.g. [6,238](this theorem also admitsformulations quite different from the one we presented, seee.g. [276, 375]). Itimplies that in Hamiltonian systems withn degrees of freedom, typically quasi-periodic Lagrangeann-tori occur with positive Liouville measure in the phasespace. As said before in§ 1.2, ‘typically’ here means that classes of examples ex-ist that areC∞-ly open. These examples are close to certain Liouville integrablesystems [12, 14, 17, 129, 297] or are locally so. For instance, this description ap-plies to any Hamiltonian system with2 or more degrees of freedom near a so-called Birkhoff nondegenerate elliptic equilibrium point. Here strong resonancesare forbidden, which by Normal Form Theory [11, 12, 17, 25, 38, 81, 90, 91, 118,120,190,313,315,371,403] implies local near-integrability. Moreover, an appro-priate local Kolmogorov nondegeneracy condition has to be satisfied, amountingto the nonvanishing of a specific normal form coefficient. As aconsequence, ina neighborhood of such an equilibrium, there are manyKAM tori. Their unionis of positive Liouville measure, where the equilibrium point even is a Lebesguedensity point of quasi-periodicity [68,346].

Iso-energetic nondegeneracy. It should be mentioned here that the above con-clusions generically also apply when restricting to the energy levels. This is aconsequence of the so-callediso-energeticKAM theorem, which is a slight varia-tion of Theorem 6. The difference is mainly due to the nondegeneracy conditionsimposed on the integrable approximation.

Indeed, for the ‘standard’KAM Theorem 6 the Kolmogorov nondegeneracycondition [6, 238] requires that the derivative of the frequency mapω : A → Rn

should have maximal rankn; this implies that the frequency mapy 7→ ω(y) locallyis a diffeomorphism.

The Arnold condition for iso-energetic nondegeneracy [6,7] similarly requiresthatω should nowhere vanish inA and the derivative of the corresponding frequency-ratio map

A 3 y 7→ [ω(y)] = [ω1(y) : ω2(y) : . . . : ωn(y)] ∈ Pn−1(R)

should have maximal rankn − 1 on each energy hypersurfaceH−1(c), wherePn−1(R) is the (n − 1)-dimensional real projective space, also see [12, 17, 65,68, 136, 223, 399, 400, 403, 421]. In coordinates, this condition means that the

33

so-called Arnold determinant

det

(∂ω/∂y ω

ω 0

)

of ordern + 1 vanishes nowhere inA. Several equivalent reformulations of theiso-energetic nondegeneracy condition are compiled in [136,400].

Russmann nondegeneracy. Another companionKAM theorem was proven byH. Russmann in the mid 1980’s. In fact, Russmann announcedhis result in [368]and presented the proof in a number of talks; a detailed written account of theproof, however, appeared only in a 1998 preprint which was published in 2001[370]—three years later still (see also [373]). Meanwhile,different proofs werepublished in the mid 1990’s by other authors [68, 106, 390, 391]. Let againTn ×A = (x, y), for A ⊆ Rn open, be the domain of definition of a Hamiltonfunction H, where integrability amounts tox-independence ofH. As before,let ω(y) = ∂H(y)/∂y be the corresponding frequency vector. We now say thatthe integrable Hamiltonian systemX = XH (as well as its frequency mapω) isRussmann nondegenerateon A, if there exists a positive integerQ such that foreachy ∈ A, the collection of all the

(n+Q

n

)partial derivatives

Dqω(y) =∂q1+···+qn

∂yq1

1 · · ·∂yqnn

ω(y)

of the frequency mapω : A → Rn at y of all the orders from0 to Q spansRn

(i.e., the linear hull of these derivatives isRn). Roughly speaking, the manifoldω(A) ⊂ Rn winds and curves enough, to have a measure-theoretically large in-tersection with the Diophantine setDτ,γ(R

n). Such varieties are studied in thetheory of Diophantine approximations on submanifolds of Euclidean spaces, alsoknown as Diophantine approximations of ‘dependent quantities’. It directly fol-lows that Kolmogorov nondegenerate or iso-energetically nondegenerate systemsare Russmann nondegenerate withQ = 1.

If A is connected andH is real analytic then Russmann nondegeneracy ofX onA is ‘almost’ equivalent to that the setω(A) does not lie in any linear hyperplanein Rn passing through the origin. To be more precise, the latter property of thefrequency mapω is equivalent to thatX is Russmann nondegenerate on any openand bounded subsetA′ ⊂ A whose closure is contained inA (with the numberQpossibly depending onA′) [67,68,368,370,447].

Russmann’s theorem states that Russmann nondegeneracy of X implies thepresence of many perturbed Diophantine quasi-periodic Lagrangeann-tori in anyHamiltonian systemX sufficiently close toX (so that the union of these tori fillsup positive measure). However, in the Russmann case, thereis in generalno

34

connection between the unperturbed and the perturbed frequencies (see a detaileddiscussion in [17,68,390,399,400]), so we cannot speak of the persistence of theunperturbed toriTn × y. Actually, Russmann nondegeneracy is much weakerthan the other two nondegeneracy conditions we have considered. For instance,the imageω(A) of the frequency map for a Russmann nondegenerate integrableHamiltonian system can be a submanifold ofRn of anypositive dimensiond 6 n,see [390, 391, 399] and [68], Example 4.7. Finally, it is worthwhile to note that ifω(A) is contained in some linear hyperplane inRn passing through the origin thenall the invariant toriTn×y can be destroyed by an arbitrarily small perturbationof X [68,390,399].

That Diophantine approximations on submanifolds of Euclidean spaces are re-quired in many problems in Mathematics and in Mathematical Physics was firstpointed out by V.I. Arnold in 1968 in his lecture “Problems ofDiophantine ap-proximations in analysis” at a symposium in the Russian cityof Vladimir (seealso a discussion in [11]). To the best of the authors’ knowledge, the first applica-tion of Diophantine approximations of dependent quantities in KAM Theory wasdue to I.O. Parasyuk [336] in 1982.

REMARKS.

1. The conclusions drawn in the example of two coupled oscillators of§ 3.2.2also directly follow as an application of the iso-energeticKAM theorem.

2. In the case of Kolmogorov nondegeneracy, the Diophantineset is pulledback along the frequency map in a locally diffeomorphic way.Arnold’s iso-energetic nondegeneracy means that the energy hypersurfacesH−1(c) aretransversal to the Diophantine half line ‘bundle’ of then-dimensional ana-logue of Figure 3. With help of these ideas, a straightforward proof of theiso-energeticKAM theorem is possible from Theorem 6 [65, 223]. The twonondegeneracy conditions—Kolmogorov and iso-energetic—are indepen-dent, as simple examples show, see e.g. [400] and [68],§ 4.2.3 (examplesfor n = 2 are also given in [136, 403]). In typical cases, however,bothnondegeneracy conditions are satisfied, implying that the union of quasi-periodic Lagrangean invariant tori has postive measure in the phase space,in such a way that the conditional measure within the energy hypersurfacesalso is positive.

3. It is also possible to derive Russmann’sKAM theorem from Theorem 6 (tobe more precise, from its analogue admitting external parameters), consid-ering the geometry and number theory of the Diophantine setDτ,γ(R

n) inmore detail [390,391], compare with§ 8.3.2 below.

35

4. All the three nondegeneracy conditions above are open in theCk-topologyfor k sufficiently large, includingk = ∞, compare with§ 4.1.2. The sameholds for the compact-open topology in the real analytic case. We like toadd here that by the Analytic Unicity Theorem, in real analytic systemscertain nondegeneracies are easily seen to be satisfied almost everywhere.

Normal ‘triviality’. In §§ 4.2.1 and 4.3.1, we dealt with integrable systems,where the invariantn-tori are normally ‘trivial’ (or even normally ‘irrelevant’),namely within center manifolds [68, 69, 223] and with the ‘classical’ case of La-grangean tori in Hamiltonian systems [6, 31, 109, 110, 184, 238, 319, 346, 349,369, 374]. A similar situation occurs in certain reversiblesystems [10, 18, 22,66, 223, 310, 312, 315, 380] or in the volume preserving case for codimension1tori [34,38,39,67–69,223,311].

4.4 Applications of the LagrangeanKAM Theorem 6

Although pure Liouville integrability is quite degenerate, integrable systems occura lot as approximations. We already met the example of a Birkhoff normal formtruncation near a nondegenerate elliptic equilibrium (seethe beginning of§ 4.3.2).In fact, we already mentioned several times that invariant Lagrangean tori withDiophantine quasi-periodic dynamics occur in Hamiltoniansystems in a typicalway. There are quite a few examples of classical, nearly integrable systems thathave received a lot of attention in the literature [4, 7, 12, 14, 17, 20, 142, 193, 194,313,315,366,403], therefore we will be restrictive here.

4.4.1 Applications in Classical, Quantum, and StatisticalMechanics

We shall briefly deal with the stability problem of the Solar System, with theAnderson localization, and with the Ergodic Hypothesis.

The Solar System. As a historical application ofKAM Theory in Classical Me-chanics, consider the Solar System, seen as a perturbation of the integrable systemobtained when neglecting the interaction between the planets. If Theorem 6 wouldapply, it would follow that the actual evolution has positive probability to be quasi-periodic, when assuming the initial conditions to have beenchosen at random. Inthat case the Solar System would be called ‘stable’. Much hasbeen said aboutthis example [4,7,17,96–98,142,160,255,278,313,315,323,357,366], and herewe just give a few remarks (see also§ 1.2).

Firstly the Solar System contains quite strong resonances [1, 7, 17], which ne-cessitate a more suitable integrable approximation than the one described here.

36

Secondly the interaction between the planets probably is far too strong for an ac-tual application of Theorem 6 as a perturbation result. The third remark refersto recent numerical work by J. Laskar [254], which seems to show that the So-lar System is entirely chaotic, mankind just does not exist long enough to havenoticed. . .

Quantum Mechanics. Other applications of theKAM techniques occur in Quan-tum Mechanics, in particular in the study of the so-called electron (Anderson) lo-calization. If one considers the Schrodinger equations with spatially ergodic po-tentials, a localized (non-conducting) state is an eigenfunction of the Schrodingeroperator for some energy eigenvalue. Such a function will decay exponentially inspace.

Since in the localized regime the Schrodinger operator typically has a densepoint spectrum, one may develop a perturbation theory of thecorresponding re-solvent operator, which diverges at this dense set of eigenvalues. These problemsare very similar to those in theKAM set-up, where the dense set of resonancesgives rise to divergent perturbation expansions.

VariousKAM -inspired proofs and analyses of localization have been proposedfor typical realizations of random potentials in arbitrarydimensions for large inter-action strengths or high energies, see e.g. [167], as well asfor quasi-periodic po-tentials in one dimension, describing electrons in quasicrystals, see e.g. [74, 143,227,257,317]. For further developments on spectral properties of the Schrodingeroperators with periodic and quasi-periodic potentials also see [70, 74, 77, 78, 154,157, 350, 367]. It turns out thatKAM Theory can be developed to show the ex-istence of a Cantor spectrum. Moreover, applications of Singularity Theory giveindications for a generic theory of gap-closing.

Yet another field of physics which is notorious for divergentperturbation the-ory problems and whereKAM -like ideas are starting to play a significant role isQuantum Field Theory [37,153,161,175].

Ergodic Hypothesis. Statistical Mechanics deals with particle systems that arelarge, often infinitely large. The taking of limits as the number of particles tendsto infinity is a notoriously difficult subject. Here we discuss a few direct conse-quences of Theorem 6 for many degrees of freedom. This discussion starts withKolmogorov’s papers [238, 239], which we now present in a slightly rephrasedform. First, we recall that for Hamiltonian systems (say, with n degrees of free-dom), typically the union of Diophantine quasi-periodic Lagrangean invariantn-tori fills up positive measure in the phase space and also in the energy hyper-surfaces. Second, such a collection ofKAM tori immediately gives rise to non-ergodicity, since it clearly implies the existence of distinct invariant sets of positive

37

measure. For background on Ergodic Theory, see e.g. [14, 24,25, 171, 172, 293].Apparently theKAM tori form an ‘obstruction’ to ergodicity, and a question is howbad this obstruction is asn → ∞. To fix thoughts we give an example.

EXAMPLE 7 (A lattice system). Consider the1-dimensional latticeZ ⊂ R, at thevertices of which identical nonlinear oscillators are situated. For simplicity thinkof the lattice being situated on a horizontal line, where at all the vertices identi-cal pendula are suspended, subject to constant vertical gravity. Also we connectnearest neighbor oscillators by weak springs, the spring constants can either bethe same for all the oscillator pairs or decay at infinity.

Let ΛN ⊂ Z be the box with vertices in the interval[−N, N ]. Then, forM 6

N consider two of these boxesΛM ⊆ ΛN . We ‘truncate’ the infinite system byignoring all the pendula outside the larger boxΛN .

First consider the integrable system associated toΛN , where all interactionsare neglected. Suppose that the oscillators situated at thevertices inΛM are inmotion, while the others are at rest. In the phase space this corresponds to aninvariant(2M + 1)-torus, which is normally elliptic (see§§ 8.2 and 8.3 below fora rigorous definition of normally elliptic invariant tori inHamiltonian systems).Moreover, the normal frequencies of this torus are in1 : 1 : . . . : 1 resonance.

Then we ‘turn on’ the activity of the interaction springs. A suitable adaptation[228] of Theorem 6 and of the results of [347] for this case yields the persistence ofthese elliptic tori for small values of the spring constants. The corresponding kindof motion is a ‘quasi-periodic breather’; for a similar typeof motion in the periodiccase see [289]. The union of elliptic tori has positive2(2M + 1)-dimensionalHausdorff measure in the phase space.

The question now is what is the asymptotics of the density of this measure asN (andM) → ∞. A partial answer to this question [228] says that this densitydecays at least exponentially fast inN , while there is a further at least polynomialdecay inM .

What conclusions can be drawn from this example? AlthoughKAM Theorygives typical counterexamples to the Ergodic Hypothesis, is seems that the corre-sponding ‘obstruction’ to ergodicity is not too bad as the size of the system tendsto infinity. This is in the same spirit as an earlier result by Arnold [8].

As we already pointed out, taking the limit asN → ∞ is an extremely difficultproblem. Another question is what happens if the limitN → ∞ is really attained.The KAM Theory for infinite systems is fully in development (see [32,233, 244–247, 314, 347] and references therein), but infinite latticesystems still have manysecrets, compare with e.g. [354].

38

4.4.2 Discussion

We conclude this section with a few remarks on the general dynamics in a neigh-borhood of HamiltonianKAM tori. In particular this concerns so-called ‘superex-ponential stickiness’ of theKAM tori and adiabatic stability of the action variables,involving the so-called Nekhoroshev estimate.

To begin with, emphasize the following difference between the casesn = 2andn > 3 in Theorem 6. Forn = 2 the level surfaces of the Hamiltonian arethree-dimensional, while the Lagrangean tori have dimension two and hence codi-mension one in the energy hypersurfaces. This means that foropen sets of initialconditions, the evolution curves are forever trapped in betweenKAM tori, as thesetori foliate over nowhere dense sets. This implies perpetual adiabatic stability ofthe action variables. In contrast, forn > 3 the Lagrangean tori have codimensionn − 1 > 1 in the energy hypersurfaces and evolution curves may escape.

REMARK . This actually occurs in the case of so-calledArnold diffusion. Theliterature on Arnold diffusion is immense, and we here just quote [8,14,108,114,116, 139, 181, 278, 280, 281, 294, 295, 299, 323, 377, 422] forresults, details, andreferences.

Next we consider the motion in a neighborhood of theKAM tori, in the casewhere the systems are real analytic or at least Gevrey smooth.

First we mention that, measured in terms of the distance to the KAM torus,nearby evolution curves generically stay nearby over asuperexponentiallylongtime [307]. This property often is referred to as ‘superexponential stickiness’ oftheKAM tori, see§ 8.1 below for more details.

Second, nearly integrable Hamiltonian systems, in terms ofthe perturbationsize, generically exhibitexponentiallylong adiabatic stability of the action vari-ables, see e.g. [11,68,136,182,185,265,278,279,281–283,294,306,308,321,323,324,332,348,377]. This property is referred to as theNekhoroshev estimateor theNekhoroshev theorem. For related work on perturbations of so-called superinte-grable systems, see [158]. The exponential stability of elliptic equilibria has beenstudied in e.g. [159,183,330].

5 Further developments inKAM Theory

In this section we deal with ParametrizedKAM Theory as this was initiated byJ.K. Moser in the 1960’s. This theory was set up in a unifying Lie algebra format,thereby covering many classes of dynamical systems characterized by preserva-tion of a certain structure, like the general ‘dissipative’case, Hamiltonian andvolume preserving cases, etc.

39

5.1 Background

We here discuss uniqueness ofKAM tori, heavily using Whitney differentiability.Also we present a few elements of the Paley–Wiener theory on Fourier series,which are fundamental for the background mathematics.

5.1.1 Unicity of KAM tori

The ‘classical’KAM Theorem 6 establishes persistence of invariant Lagrangeantori in nearly integrable Hamiltonian systems. These tori are quasi-periodic withDiophantine frequency vectors and their union is a nowhere dense set of positivemeasure in the phase space. It is a long standing question of how far the perturbedtori are unique. Using the fact that at the level of the tori, there exists a Whitneysmooth conjugation between the integrable approximation and its perturbation,this unicity follows on a closed subset of the Diophantine torus union of full mea-sure [83] (see also [374]). We first introduce this subsetDd

τ,γ(Rn) ⊆ Dτ,γ(R

n).To explain this, in general letK ⊆ Rn be a closed set. We say thata ∈ K is

a density pointprecisely if anyC∞-functionF : Rn → R, such thatF |K ≡ 0,has an infinite-jetj∞(F )(a) = 0. The set of all density points ofK is denoted byKd. Moreover, in general we say that the closed setK ⊆ Rn possesses the closedhalf line property if the following holds: wheneverp ∈ K ands > 1, then alsosp ∈ K.

LEMMA 8 (Properties ofKd) [83]. LetK ⊆ Rn be a closed set. Then

1. Kd ⊆ K is a closed set;

2. K \ Kd has Lebesgue measure zero;

3. If K possesses the closed half line property, then so doesKd.

The proof is rather direct, item 2 using the Fubini theorem. Applying this con-struction toK = Dτ,γ(R

n) gives the subsetDdτ,γ(R

n) ⊆ Dτ,γ(Rn). Recall from

§ 4.1.2 thatDτ,γ(Rn) and henceDd

τ,γ(Rn) possess the closed half line property.

THEOREM 9 (Unicity of KAM tori) [83]. Assuming the set-up of the ‘classical’KAM Theorem 6, there existC∞-neighborhoodsU2 of XH andV2 of the identitymapIdM , such that the following holds. IfΦ ∈ V2, restricted toTn × Dτ,γ(Aγ),is a conjugation between the vector fieldsXH andXH+F ∈ U2, then the furtherrestriction ofΦ, to Tn × Dd

τ,γ(Aγ), is unique up to a torus translation.

REMARKS.

40

1. We note that the above definition of a density point does notcoincide withthat of a Lebesgue density point. A general problem is to characterizeDd

τ,γ(Rn) ⊆ Dτ,γ(R

n), compare with§ 4.2.1.

2. Again the somewhat smaller closed half line structure ofXH is inherited bythe perturbationXH+F , up to a diffeomorphism.

3. It is conjectured [83] that Theorem 9 generalizes to certain otherKAM the-orems as well, e.g., to Theorem 3.

5.1.2 Paley–Wiener estimates and Diophantine frequencies

As before, letTn = Rn/(2πZ)n be the standardn-torus with coordinatesx1, . . . , xn

counted modulo2πZ. We consider functionsh : Tn → R of classCr, for r > 0.One of the main tools in the proofs of theKAM theorems is the solution of linear(or homological) partial differential equations of the form

n∑

j=1

ωj∂h(x)

∂xj= H(x), (22)

whereH : Tn → R is given withTn-average0. These equations have to be solvedfor h, also compare with§ 3.1.1. In the proofs of all theKAM theorems for flows,as discussed here, this linear problem is central in a Newtonian iteration processthat solves a nonlinear conjugation equation under Diophantine conditions. Wenow discuss this problem in terms of Fourier series, for details referring to, e.g.,[68,83,123,346].

Fork ∈ Zn thekth Fourier coefficient ofh is given by

hk =1

(2π)n

∮

Tn

e−i〈k,x〉h(x) dx, (23)

and we consider the formal Fourier series

h(x) =∑

k∈Zn

hk ei〈k,x〉.

We have the following familiar norms in terms of the Fourier coefficients:

|||h|||∞ = maxk

|hk|;

|||h|||2 =

(∑

k

|hk|2)1/2

;

|||h|||1 =∑

k

|hk|.

41

For any (continuous) functionh : Tn → R we recall that

|||h|||∞ 6 |||h|||2 6 ‖h‖0 6 |||h|||1. (24)

Since we did not require the Fourier series to converge, someof these norms maybe infinite. For any continuous functionh : Tn → R we decompose

h = h0 + h,

with h0 as in (23), i.e., withh0 equal to theTn-average ofh. We callh thevariablepart of h.

LEMMA 10 (Paley–Wiener estimates).Leth : Tn → R be of classCr, with vari-able parth.

1. Then there exists a positive constantC(1)r,n such that for allk ∈ Zn \ 0

|k|r|hk| 6 C(1)r,n

∥∥h∥∥

r.

2. Moreover, forr > n + 1 there exists a positive constantC(2)r,n such that

∥∥h∥∥

06 C(2)

r,n maxk 6=0

|k|r|hk|.

REMARKS.

1. We mention that the first item of Lemma 10 is the familiar Paley–Wienerdecay for the Fourier coefficients of aCr-function, which directly followsby partial integration. For the second item of the lemma, it suffices to takeC

(2)r,n =

∑k 6=0 1

/|k|r.

2. We conclude that ifh is of classC∞, its Fourier coefficients decay fasterthan any positive power of|k|−1. Similarly, whenh is real analytic thisdecay is exponentially fast.

LEMMA 11 (Small divisors).Assume thath : Tn → R satisfies the differentialequation(22) for a givenH : Tn → R of classCr with Tn-average0, whereω ∈ Dτ,γ(R

n), see(18). Then:

1. For all k ∈ Zn \ 0 one has that

|hk| 6|k|τγ

|Hk|.

42

2. Moreover, forr > n+ p+ τ with p ∈ N one has thath ∈ Cp−1(Tn, R) with

∥∥h∥∥

p−16

C(3)r,n,p,τ

γ‖H‖r,

for a positive constantC(3)r,n,p,τ .

REMARKS.

1. The first estimate follows by comparing terms in the Fourier series, usingthe Diophantine condition (18). To illustrate the next estimate we considerthe casep = 1. It follows from the last inequality in (24) and Lemma 10that for theC0-norm ofh

∥∥h∥∥

06∣∣∣∣∣∣h∣∣∣∣∣∣

16∑

k 6=0

|k|τγ

|Hk|

6∑

k 6=0

|k|τγ

|k|−r C(1)r,n‖H‖r 6

C

γ

∑

`>1

`τ−r‖H‖r `n−1

= C‖H‖r

γ

∑

`>1

`n+τ−r−1 = C ′‖H‖r

γ

(` = |k|), where we have used thatn + τ − r − 1 6 −2, which givesconvergence of the last sum.

2. We conclude that ifH is of classC∞, then so ish. Similarly wheneverHis real analytic, then so ish.

5.2 ParametrizedKAM Theory

Already in§§ 3.1 and 4.2 we met parameters inKAM Theory, which were neededto get persistence of quasi-periodic invariant tori of the integrable approximation.Below we shall generalize this approach by developing ParametrizedKAM Theoryin a more systematic way and taking the normal linear dynamics into account.

In [311, 312], J.K. Moser presented two directions in which one may general-ize the set-up of Section 4, that are logically connected. Firstly, it turns out thatKAM Theorem 6 can be directly carried over to the reversible setting, to the gen-eral dissipative setting, etc., compare with Theorem 5 and see [69,223,315,380].These cases are characterized by the fact that they are normally ‘trivial’ or ‘irrel-evant’, see the end of§ 4.3.2. Secondly, the possibly nontrivial normal behaviorof the invariant tori can be taken into account within a ‘modifying term’ formal-ism. This means that, in order to obtain a conjugation between the unperturbed

43

and the perturbed tori, the system has to be changed ‘at lowest order’, a suitablenondegeneracy condition having to be fulfilled. This resultgives, e.g., an alter-native approach toKAM Theory of lower dimensional isotropic invariant tori inHamiltonian systems (see§ 8.3). Here, apart from the internal frequencies ofthe quasi-periodic tori, also the normal frequencies play arole and enter the Dio-phantine conditions. The latter procedure turns out to allow also for a furthergeneralization of the non-Hamiltonian settings mentionedabove. In fact, thereexists a quite general formulation of this theory in terms ofLie algebras of vectorfields, encompassing the Hamiltonian, volume preserving, and several equivariantcases, etc., as well as the reversible set-up. For an axiomatic approach to these‘admissible’ Lie algebras, see [62,69,216,223,312], the reversible counterpart istreated in [49,66,121].

REMARK . Compare this with the Formal Normal Form Theory at equilibria,periodic solutions, and quasi-periodic tori as developed in a structure preservingLie algebra formalism [38, 41], that similarly also covers the dissipative normalforms [11, 22, 90–92, 118, 120, 189, 249, 440], the Hamiltonian (Birkhoff) normalforms [12,17,25,90,91,118,120,190,313,371,372,403], etc.

This line of thought was picked up in [67–69, 223] as follows.Instead of con-sidering a given system that has to be modifiedlater, the theorystarts off withfamilies of systems, i.e., with the set-up where the systemsdepend on (external)parameters. This determines a fixed universe of parametrized systems that can beeither integrable or not. Generally integrability is defined as invariance under thenatural action of a torus group, where the invariant tori in question are orbits ofthe action, compare with§ 4.2.1. It turns out that Moser’s nondegeneracy con-dition now translates to (trans)versality of the integrable family, to be called theBHT (Broer–Huitema–Takens) nondegeneracy. Moreover, perturbations of thisfamily contain the perturbed tori, for which the connecting, parameter dependentconjugation preserves not only the internal frequencies ofthe tori, but their entirenormal linear part as well. Furthermore, the (trans)versality of the integrable fam-ily also affects this normal linear part, which now takes, toa large extent, the roleof the ‘lowest order’ terms mentioned above.

5.2.1 The parametrized dissipativeKAM theorem

For simplicity we consider in this section the following setting. As the phase spacewe takeM = Tn × Rm = (x, y), and as the parameter spaceP ⊆ Rs = µ,an open subset. The starting point is an integrableC∞-family X = Xµ(x, y) ofvector fields, given by

x = ω(µ) + O(|y|)y = Ω(µ)y + O(|y|2), (25)

44

where theO-estimates are (locally) uniform inµ. Hereω(µ) ∈ Rn andΩ(µ) ∈gl(m, R) for each value ofµ. Integrability ofX again amounts tox-independence,compare with§ 4.2.1. The interest is with persistence properties of the familyTµ = Tn × 0 of invariantn-tori with parallel dynamics. The local nondegen-eracy condition roughly means that the pair of maps consisting of the internalfrequency vectorω(µ) and the ‘normal’ matrixΩ(µ), nearµ = µ0 has to varysufficiently withµ ∈ P . In particular, the mapµ 7→ ω(µ) is a submersion as in§ 4.2.1, while at the same time the mapµ 7→ Ω(µ) is aversal unfoldingof Ω(µ0),so it is transversal to the orbit ofΩ(µ0) under the adjoint action ofGL(m, R).Note that versal unfoldings with a minimal number of parameters are said to beminiversal, compare with [9, 11], also see e.g. [180]. In coordinates the normallinear part of (25) is given by

ω(µ)∂

∂x+ Ω(µ)y

∂

∂y. (26)

For a coordinate free definition of the normal linear part of integrable systems atinvariant tori, using the normal bundle of the tori, see [69]I § 2 and [223]§ 2.

BHT nondegeneracy. We shall confine ourselves to the case whereΩ has onlysimple eigenvalues. Suppose that these eigenvalues are given by

(δ1, . . . , δN1, α1 ± iβ1, . . . , αN2 ± iβN2

)(27)

with βj > 0 for 1 6 j 6 N2. Note thatN1 + 2N2 = m. We call β =(β1, β2, . . . , βN2) the normal frequenciesof the invariant torus. Next considerthe map

spec : gl(m, R) → RN1 × RN2 × RN2 ; Ω 7→ (δ, α, β), (28)

which by simpleness of the eigenvalues parametrizes theGL(m, R) orbit spacenearΩ(µ0). The BHT nondegeneracy condition in this case boils down to sayingthat atµ = µ0 the map

P 3 µ 7→(ω × (spec Ω)

)(µ) ∈ Rn × RN1 × RN2 × RN2 (29)

is a submersion [67–69, 223]. As in§ 4.2.1, we can use the Inverse FunctionTheorem [214,318,407] to reparametrize

µ ↔ (ω, δ, α, β)

on an open subsetA ⊆ P , suppressing for simplicity extra parameters that maypossibly occur. We then say that the familyXµ is nondegenerate on the torusunionTn × 0 × A =

⋃µ∈A Tµ, recall that here0 ∈ Rm.

45

Diophantine conditions. Now we need Diophantine conditions on the internaland normal frequencies. Forτ > n − 1 and γ > 0 define the set of(τ, γ)-Diophantine normal-internal frequency vectors by

Dτ,γ(Rn; RN2) =

(ω, β) ∈ Rn × RN2

∣∣ |〈ω, k〉 + 〈β, `〉| > γ|k|−τ , (30)

for all k ∈ Zn \ 0 and for all` ∈ ZN2 with |`| 6 2,

compare with (18). This set is again a nowhere dense set of positive measure(for γ sufficiently small) with the closed half line property, see§§ 4.1.2 and 5.1.1;compare with [67–69,223,311,312]. DefiningΓ =

(ω× (spec Ω)

)(A), without

any restriction we may assume thatΓ has the ‘product form’Γ = Γω ×Γδ ×Γα ×Γβ. Furthermore we define the shrunken version

Γγ =(ω, δ, α, β) ∈ Γ

∣∣ dist((ω, δ, α, β), ∂Γ

)> γ

of Γ as well as

Dτ,γ(Γγ) = Γγ

⋂(Dτ,γ(R

n; RN2) × Γδ × Γα

)

andDτ,γ(Aγ) ⊂ A, compare with§§ 4.2.1 and 4.3.1. Note that the closed half linesof Dτ,γ(R

n; RN2) now turn into closed linear half spaces of dimension1+N1+N2.Again these geometrical structures, up to a diffeomorphism, are inherited by theperturbations. This is a consequence of the following theorem.

THEOREM 12 (ParametrizedKAM – dissipative case) [62,69,123,216,223].Letn >

2. Let the integrableC∞-familyX = Xµ(x, y) of vector fields(25) beBHT non-degenerate onTn × 0 × A, with A ⊆ P open. Also assume that forµ ∈ Awe havedet Ω(µ) 6= 0. Then, forγ > 0 sufficiently small, there exists a neigh-borhoodO of X in theC∞-topology, such that for any perturbed (not necessarilyintegrable) familyX ∈ O there exists aC∞-mappingΦ : Tn × Rm × A →Tn × Rm × A, defined nearTn × 0 × A, with the following properties:

1. Φ is a C∞-near the identity map preserving projections to the parameterspaceP .

2. The image of theX-invariant torus unionTn × 0 × Dτ,γ(Aγ) underΦ isX-invariant, and the restricted mapΦ = Φ |Tn×0×Dτ,γ(Aγ) conjugatesXto X, that is

Φ∗X = X.

46

3. Φ preserves the normal linear behavior of the toriTµ for µ ∈ Dτ,γ(Aγ) withrespect toX. This means the following. Let

Φ(x, y, µ) =(Ψ[µ](x, y), Υ[µ]

),

whereΨ[µ](x, y) ∈ Tn × Rm andΥ[µ] ∈ P (that theµ-componentΥ[µ] ofΦ does not depend on the phase space variablesx andy just expresses thepreservation of projections to theµ-space). Then for eachµ ∈ Dτ,γ(Aγ) thevector field (

Ψ[µ])−1

∗XΥ[µ]

is given by(25), where now theO-terms are, generally speaking,x-depending.

REMARKS.

1. The conclusion of Theorem 12 first of all expresses that thefamily X isquasi-periodically stable on the union

⋃

µ∈Dτ,γ(Aγ)

Tµ = Tn × 0 × Dτ,γ(Aγ)

of Diophantine quasi-periodic invariantn-tori [67–69, 223], compare with§§ 3.1.3, 4.2.1, 4.3.1 above. Including item 3 of Theorem 12, wealso speakof normal linear stabilityof this torus union.

2. Theorem 12 was also stated and proven in the more general setting withpreservation of certain structures. This includes, next tothe general dissipa-tive case of Theorem 12 itself, the Hamiltonian and the volume preservingcases, as well as certain equivariant and reversible cases,all of these withexternal parameters if necessary. In fact, the parametrized KAM theoremwas stated and proven for certain ‘admissible’ Lie algebrasof vector fieldsand for reversible analogues of such algebras, as we alreadymentioned atthe beginning of§ 5.2, for an axiomatic approach see [49,62,66,69,121,216,223,312]. In many set-ups, in the definition of the mapspec we have to re-frain from counting double. This refers for instance to the Hamiltonian andreversible settings, where complex eigenvalues can show upin quadrupletswhile non-zero real eigenvalues come in pairs (see§ 8.3.1). For an elaboratediscussion see [69] I§ 2, [223]§ 2, and [66]. The Hamiltonian counterpartof Theorem 12 will be presented in§ 8.3.1 below, see Theorem 17.

3. The assumption thatΩ(µ) has only simple eigenvalues for eachµ can bedropped, even in the full generality of the structure preserving settings men-tioned in the previous item. Here we use the fact that Arnold’s theory of

47

matrices depending on parameters [9, 11] can be carried overto a largeclass of ‘admissible’ Lie subalgebras ofgl(m, R) under the adjoint actionof the corresponding subgroups ofGL(m, R) and to reversible analoguesof such actions. In this general case versal unfoldings can be normalized tothe so-calledlinear centralizer unfolding, for definitions and other math-ematical details see e.g. [9, 11, 118, 168, 180, 237, 341]. Note that thisdefinition encompasses the above case of simple eigenvalues. Also com-pare with [84, 335] concerning genericity in terms of transversality. Thecorresponding ParametrizedKAM Theory has been worked out in detailin [48, 49, 54, 62, 121, 122, 216] for Hamiltonian and reversible variants ofthe normal1 : −1 resonance.

4. In the above set-up the condition thatdet Ω(µ) 6= 0 cannot be omitted, al-though it was not needed in [311,312]. This problem recentlywas overcomeby Wagener [432,433].

5.2.2 Direct consequences of the parametrized approach

Partly recalling [67–69, 223], we briefly discuss here a number of straightfor-ward consequences of the parametrized approach. First of all, the normally ‘triv-ial’ cases (see the end of§ 4.3.2) without external parameters, like the case ofLagrangean tori in the Hamiltonian setting (Theorem 6), thecase of codimen-sion 1 tori in the volume preserving setting, as well as the ‘standard’ reversibleset-up [10,18,22,223,312,315,380], directly follow using so-calledlocalization,see [69] I§ 5 and [223]§ 5 (compare also with Remark 2 after Theorem 17 be-low). This means that by introducing an extra local multi-parameterµ we end upexamining a familyTµ of invariant tori as before, where we have to consideronlyone torusfor each value ofµ. Moreover, sinceΩ(µ) ≡ 0, we only need the con-stant unfolding in this direction. After application of thecorresponding versionof ParametrizedKAM Theory, we can project back to obtain a persistence resultwithout parameters. In particular this holds for Theorem 6,which was actuallyalso proven in this way [312, 349]. Also the center manifold situation of Theo-rem 5 can be seen as a particular case of parametrizedKAM Theorem 12, for amore elaborate discussion see [69] I§ 7 and [223]§ 7.

REMARKS.

1. As explained in [69] I§ 7 and [223]§ 7, all cases with varying frequencyratios [ω1 : ω2 : . . . : ωn] ∈ Pn−1(R) fall under this approach, leading toweak quasi-periodic stability. Indeed, a scaling of the time gives an extraparameter, after which ParametrizedKAM Theory applies. After project-

48

ing to the original setting, the conjugationΦ turns into an equivalence, fordefinitions see e.g. [335].

2. A similar discussion applies to the Hamiltonian iso-energetic setting [12,17,65,136,223,399,400,421], for a geometric discussion see§ 4.3.2 above.

3. The set-up of ParametrizedKAM Theory is particularly suitable for studyingquasi-periodic bifurcations. In the next two sections we shall come back tothis subject.

Russmann nondegeneracy revisited. Preservation of the normal linear behav-ior surely may require a lot of external parameters, and a clear aim always is to getrid of as many parameters as possible, compare with [69] I§ 7c and [223]§ 7c,using the geometry of the Diophantine sets (see the previousdiscussion as an ex-ample). In fact, Theorem 12 and all its structure preservinganalogues possess‘miniparameter’ versions [67, 68] with Russmann-like nondegeneracy conditionson the unperturbed frequency maps. These conditions are slight generalizations ofthe Russmann nondegeneracy condition of§ 4.3.2 to the case where the frequencymap depends on external parameters. The ‘miniparameter’KAM theorems can bemost easily obtained using the so-called Herman method [67,68, 389–391, 396,400, 401]. In§ 8.3.2, we shall illustrate Herman’s approach for the Hamiltoniancounterpart of Theorem 12.

5.2.3 Reducibility issues

The present setting assumes that the integrable familyX has the form (25) onthe phase spaceM = Tn × Rm, while its normal linear part has the form (26).This means that the system is in Floquet form (the coefficients of the variationalequation along each invariant torus do not depend on the point on this torus). Ofcourse, the results also hold for all the cases reducible to this form.

First of all we observe that it is an immediate consequence ofparametrizedKAM Theorem 12, that reducibility is a persistent property on Diophantine sets ofparameters, compare with [69] I§ 7 and [223]§ 7.

However, it is known that this reducibility is not always possible, see e.g. [151,154,206,207,240,241]. In [82,85,88,417,429], also compare with [123], the skewHopf bifurcation was treated as a first example of non-reducibleKAM Theory. Formore details see§ 6.2.3 below. Concerning non-reducibility in the Hamiltonianlower dimensional context, see§ 8.3.3 below.

49

6 Quasi-periodic bifurcations: dissipative setting

The bifurcation theory of equilibrium points is widely developed in the generaldissipative setting, see, e.g., [11, 13, 118, 189, 249, 325,440]. As generic codi-mension1 bifurcations we mention the saddle-node (or fold) bifurcation and theHopf bifurcation [218] (also called the Poincare–Andronov phenomenon [11]).This theory has a direct extension to bifurcations of periodic solutions. The Hopfbifurcation then translates to the Hopf–Neımark–Sacker bifurcation, which bythe Poincare map relates to the Hopf bifurcation for fixed points of diffeomor-phisms [415], for a discussion also see [123]. Here normal-internal resonancesalready play a role in the interaction between the (internal) frequency of the pe-riodic solution and the normal frequency, which gives rise to a pattern of Arnoldresonance tongues comparable to Figure 2. In order to see this pattern, one needsto keep track of the normal frequency, which requires more parameters. It turnsout that the Arnold family of circle maps (11) mentioned in§ 3.1.3 to some extentis a good model for this Hopf–Neımark–Sacker scenario. Forperiodic solutionsalso the period doubling bifurcation occurs as a generic codimension1 bifurcation.We note that saddle-node, period doubling, and Hopf bifurcations all are relatedto loss of (normal) hyperbolicity. Presently the interest is with the analogue of thistheory in the case of quasi-periodic tori, compare with [35,40,43,68,69,123].

To study this we return to the phase spaceM = Tn×Rm =(

x (mod2π), y)

,where we considerC∞-familiesX of vector fields of the general form

Xµ(x, y) = Fµ(x, y)∂

∂x+ Gµ(x, y)

∂

∂y

(µ being a suitable multi-parameter). As before, integrability translates to aTn-symmetry, which simply means that the coefficientsFµ andGµ arex-independent.

If X is such an integrable family, then, by dividing out theTn-symmetry, wecan reduce to aC∞-family Xred,µ of vector fields

Xred,µ(y) = Gµ(y)∂

∂y

on Rm. Notice that equilibria ofXred correspond to invariantn-tori of the inte-grable familyX; this is why these equilibria are calledrelative. Similar remarksgo for (relative) periodic solutions ofXred. In the reduced familyXred we canmeet the bifurcations described above. For the integrable family X this gives adirect translation in terms of torus bifurcations.

The problem addressed in this section is what happens to the integrable bi-furcations when perturbing to nearly integrable familiesX = Xµ(x, y). It turnsout [69] that all the three cases (saddle-node, period doubling, and Hopf) lead to

50

αααααααααααααααααααααααααααααααααααα

ββββββββββββββββββββββββββββββββββββ

Figure 4: Parametrized section of the Diophantine setDτ,γ(Γγ), whereω ∈ Rn hasa fixed Diophantine value. The Hopf lineα = 0 is Cantorized by normal-internalresonances.

typical (i.e.,C∞-open)quasi-periodic bifurcationscenarios. This results from acombination ofKAM Theory and Bifurcation Theory (and Singularity Theory) asthis goes back to Whitney, Thom, Mather, and Arnold, see e.g.[11,13,15,16,418].Generally speaking, it turns out that the (real) semi-algebraic stratifications thatoccur as bifurcation sets in the product of the phase space and the parameter space,are ‘Cantorized’ in a systematic way, compare with Figure 4.

TheKAM Theory of quasi-periodic versions of period doubling and Hopf bifur-cations is a direct application of Theorem 12, while the quasi-periodic saddle-nodebifurcation is more involved, see [69] II§ 5 and [433]. An extension regardingthe so-called quasi-periodicd-fold degenerate bifurcation, based on the so-called‘translated torus’ theorem [205,208,365,449], was made in[432]; for an applica-tion to quasi-periodicity in planar maps, see [72].

We shall illustrate the general approach on the quasi-periodic Hopf bifurcation,compare with [35,42,45,68,69,123].

51

6.1 Quasi-periodic Hopf bifurcation

The unperturbed, integrableC∞-family X = Xµ(x, y) on Tn × R2 has the form(25)

Xµ(x, y) =[ω(µ) + f(y, µ)

] ∂

∂x+[Ω(µ)y + g(y, µ)

] ∂

∂y, (31)

with f = O(|y|) andg = O(|y|2). Moreover,µ ∈ P is a multi-parameter withP ⊆ Rs open, whileω : P → Rn andΩ : P → gl(2, R) are smooth maps.Without loss of generality we may assume thatΩ has the form

Ω(µ) =

(α(µ) −β(µ)β(µ) α(µ)

).

REMARK . Note that forα 6= 0, we return to the normally hyperbolic situationof Theorem 5 of§ 4.2.1, but here we are particularly interested in the transitionoccurring whenα passes through0. Besides, dissipative parametrizedKAM The-orem 12 of§ 5.2.1 is applicable to small perturbations of (31) for anyα (withm = 2, N1 = 0, N2 = 1).

The assumption of BHT nondegeneracy leads to the existence of an open subsetA ⊆ P on which the map

P 3 µ 7→(ω × (spec Ω)

)(µ) =

(ω(µ), α(µ), β(µ)

)∈ Rn × R2

is a submersion [69], compare with (29). As in§ 5.2.1, we reparametrize

A 3 µ ↔ (ω, α, β) ∈ Γ ⊂ (Rn × R2),

suppressing for simplicity the possible occurrence of other parameters. Here wedenote byΓ the open(ω, α, β)-parameter domain, without loss of generality tak-ing it of the product formΓ = Γω × Γα × Γβ, as in§ 5.2.1, and assuring that0 ∈ Γα.

We furthermore assume that the reduced system

Xred,(α,β)(y) =[Ω(α, β)y + g(y, α, β)

] ∂

∂y(32)

exhibits a standard supercritical Hopf bifurcation [11, 13, 118, 189, 249, 440] atα = 0. This means that the (relative) equilibriumy = 0 is attracting forα < 0and repelling forα > 0, while for α > 0 a (relative) periodic solution branchesoff that is attracting, also compare with [84]. These statements hold for ‘a half’of all nonlinearitiesg: depending on the sign of the coefficient of a certain third

52

order term in the normal form forg, the system (32) admits either an attracting pe-riodic solution forα > 0 (a supercritical Hopf bifurcation) or a repelling periodicsolution forα < 0 (a subcritical Hopf bifurcation).

For the integrable familyX = Xω,α,β(x, y) this scenario directly translatesto n- and (n + 1)-tori with parallel dynamics, and the question is what is theirfate when in theC∞-topology we perturbX to a nearly integrable familyX =Xω,α,β(x, y).

6.1.1 Persistent quasi-periodicn-tori

We start answering the question of persistent invariantn-tori by applying Theo-rem 12 in the present setting. Therefore, forτ > n − 1 andγ > 0, as a specialcase of (30) we consider the set

Dτ,γ(Rn; R) =

(ω, β) ∈ Rn × R∣∣ |〈ω, k〉+ β`| > γ|k|−τ , (33)

for all k ∈ Zn \ 0 and for all` ∈ Z with |`| 6 2.

The setsΓγ andDτ,γ(Γγ) are now defined as in§ 5.2.1. For a sketch of a section ofDτ,γ(Γγ) for fixed Diophantineω, see Figure 4, compare with [35, 410]. We alsointroduce the full measure subset of density pointsDd

τ,γ(Rn; R) ⊆ Dτ,γ(R

n; R),see§ 5.1.1. We takeγ > 0 sufficiently small, so that the projectionΓαγ of Γγ

on Γα hasα = 0 as an interior point. Also we takeγ > 0 sufficiently small forthe nowhere dense setDτ,γ(Γγ) to have positive measure. Besides, we considerDd

τ,γ(Γγ), obtained by taking the product with the intervalΓαγ . As a consequenceof Theorem 12, for any familyX on Tn × R2 × P , sufficiently close toX in theC∞-topology, a near-identityC∞-diffeomorphismΦ : Tn×R2×Γ → Tn×R2×Γexists, defined nearTn ×0× Γ, that conjugatesX to X when further restrictedto Tn × 0 × Dτ,γ(Γγ).

Next consider the perturbed familyX in the coordinates provided by the in-verseΦ−1. In other words, we study the pull-back vector fieldΦ∗X = (Φ−1)∗X,that on the nowhere dense setTn × 0 × Dτ,γ(Γγ) coincides with the integrablefamily X. We directly conclude thatΦ∗X for parameter values inDτ,γ(Γγ) hasTn × 0 as a quasi-periodic invariantn-torus; this torus is attracting forα < 0and repelling forα > 0. As in § 5.1.1, we further restrict to the set of densitypointsDd

τ,γ(Γγ) ⊂ Dτ,γ(Γγ) as a full measure subset, which leads to the normalform decomposition

(Φ∗X − X

)ω,α,β

(x, y) = O(|y|) ∂

∂x+ O(|y|2) ∂

∂y+ Qω,α,β(x, y), (34)

asy → 0. For boundedΓ theO-estimates are uniform inx andω, α, β. TheC∞-family of vector fieldsQ is uniformlyflat onTn ×∆×Dd

τ,γ(Γγ) ⊂ Tn ×R2 ×Γ,

53

where∆ is a small neighborhood of0 ∈ R2. This means that the Taylor series ofQ completely vanishes. Indeed, for∆ small we can arrange thatQ vanishes ontheTn ×∆ ×Dd

τ,γ(Γγ), which by the definition of a density point implies that onthe open set all the derivatives ofQ vanish. This is what the flatness expresses.

6.1.2 Fattening the parameter domain of invariantn-tori

We keep studying the perturbed systemX in its pull-back form, so we are stillconsideringΦ∗X. Forα 6= 0, the invariantn-tori in question are normally hyper-bolic. By the Center Manifold Theorem [117,162,215,424] weconclude that theparameter domain insideΓ, where invariantn-tori exist, contains a neighborhoodof the parameter values corresponding to the Diophantine quasi-periodic tori. Inother words, the nowhere dense parameter domainDd

τ,γ(Γγ) of Φ∗X-invariantn-tori, for α 6= 0, can befattenedto an opensubset ofΓ. We note that outsideDd

τ,γ(Γγ), the dynamics on invariantn-tori doesnot have to be conditionally peri-odic.

The fattening by hyperbolicity can be carried out using a more or less well-known contraction principle, see e.g. [117], for a detailedconstruction using avariation of constants operator see [35]. We here restrict to describing the resultof the fattening operation. To this purpose we proceed as follows.

1. As before assume thatΓ = Γω ×Γα×Γβ , i.e., thatΓ is of the product form,compare with Figure 4.

2. In the frequency spaceRn \ 0 = ω, defineω = ω/|ω| ∈ Sn−1 ⊂ Rn.Also, let% : Sn−1 × Sn−1 → R+ be the metricSn−1 inherits fromRn.

3. Finally consider any monotonically increasingC∞-functionp : R+ → R+

that is (infinitely) flat at 0.

For any fixedω0 = |ω0|ω0 ∈ Γω andβ0 ∈ Γβ , such that(ω0, α, β0) ∈ Dτ,γ(Γγ)for all α ∈ Γαγ , consider sets of the form

(ω, α, β) ∈ Γ

∣∣ 0 < |α| < C andp(%(ω, ω0) + |β − β0|

)< D|α|K

, (35)

whereC, D, andK are positive constants. Notice that this is the union of twoopen discsAω0,β0 (occurring forα < 0) andRω0,β0 (occurring forα > 0), eachwith a piecewise smooth boundary, that atβ = β0 have an infinite order of contactwith the bifurcation hyperplaneα = 0.9

9In [35, 69], for historical reasons, instead of ‘disc’ the term ‘(blunt or flat conic) cusp’ wasused.

54

α

β

H

(a)

Hc

hyperbolichyperbolic

α

β

H

Hc

(b)

σ1

σ2

Aσ1

Aσ2

Rσ1

Rσ2

Figure 5: Sketch of a ‘frayed’ Hopf boundary, whereHc ⊂ H is nowhere denseof positive measure, whileH is a smooth curve. (a). Global impression of the do-mains with attracting or repellingn- and(n+1)-tori, compare with [35]. (b). For-mation of one resonance ‘bubble’ in between discs attached to σ1, σ2 ∈ Hc.

PROPOSITION13 (Fattened domain ofn-tori) [35,69]. In the above situation, givenr ∈ N, there exist positive constantsC andD with the following property. Foreach(ω0, β0) such that(ω0, α, β0) ∈ Dd

τ,γ(Γγ) for all α ∈ Γαγ , the correspondingdiscsAω0,β0 and Rω0,β0 (35) with K = 3 are contained in the parameter do-main with normally hyperbolicΦ∗X-invariant n-tori of classCr. These tori areattracting inAω0,β0 and repelling inRω0,β0.

REMARKS.

1. The discsAω0,β0 andRω0,β0 become larger as the degree of differentiabilityr decreases.

2. The domain of invariant tori is an uncountable union of discs, leaving open acountable number of ‘bubbles’ centered around the pure resonances(ω, 0, β) ∈Γ with 〈ω, k〉 + β` = 0 for somek ∈ Zn \ 0 and` = −2,−1, 0, 1, 2.

3. The resonant dynamics inside certain bubbles of the quasi-periodic Hopf bi-furcation also has been widely studied, see, e.g., [19,71,176,250,376,430].For similar studies related to the quasi-periodic saddle-node bifurcation,see [99–101].

6.1.3 The parameter domain of invariant(n + 1)-tori

In order to find invariant(n + 1)-tori, we first develop a new pull-back of the per-turbed systemX that has aTn+1-symmetric normal form truncation and which is

55

related to both the planar supercritical Hopf familyXred and to the quasi-periodicnormal form (34). To this purpose, givenN ∈ N, consider the subset

Eτ,γ;N(Rn; R) =(ω, β) ∈ Rn × R

∣∣ |〈ω, k〉 + β`| > γ|k|−τ , (36)

for all k ∈ Zn \ 0 and for all` ∈ Z with |`| 6 N,

compare with (33), which again is a nowhere dense set of positive measure (forγsufficiently small), with the closed half line property. Note thatEτ,γ;2(R

n; R) =Dτ,γ(R

n; R). Accordingly one may defineEτ,γ;N(Γγ) andEdτ,γ;N(Γγ). In these

circumstances we can roughly paraphrase Theorem 12 as follows. Forα suf-ficiently small, there exists a near-identityC∞-diffeomorphismΦ defined nearTn ×0×Γ ⊂ Tn ×R2×Γ, such that the following normal form decompositionholds:

(Φ∗X

)ω,α,β

(x, y) =

[ω + |y|2 f(|y|2, ω, α, β) + O(|y|N)

] ∂

∂x+

[β + |y|2 g(|y|2, ω, α, β) + O(|y|N+1)

] [−y2

∂

∂y1

+ y1∂

∂y2

]+ (37)

[α + |y|2 h(|y|2, ω, α, β) + O(|y|N+1)

] [y1

∂

∂y1+ y2

∂

∂y2

]+

Q(x, y, ω, α, β),

where the familyQ of vector fields is uniformly flat onTn × 0 × Eτ,γ;N(Γγ).Indeed, decomposition (37) forN > 2 is obtained by initially applying The-orem 12, followed by a standard formal normal form procedureas developedin [35,38,69,118,416]. ForN = 2 we recover (34). Thus, theTn+1-symmetry ofthe normal linear part forα = 0 is pushed over the formal series iny.

In our application we takeN = 7. Note that then the(∂/∂y)-component of(37) is close to the standard planar Hopf normal form [38,249,416]. The invariant(n + 1)-tori now can be found in a straightforward manner.

PROPOSITION14 (Fattened domain of(n + 1)-tori) [69]. In the above situation,givenr ∈ N, there exist positive constantsC andD with the following property.For each(ω0, β0) such that(ω0, α, β0) ∈ Ed

τ,γ;7(Γγ) for all α ∈ Γαγ , the corre-sponding discRω0,β0, i.e., the set(35) for α > 0, withK = 7/2 is contained in theparameter domain with normally hyperbolicΦ∗X-invariant (n + 1)-tori of classCr. These tori are attracting.

Mutatis mutandis, the same remarks apply as those followingProposition 13.

56

6.2 Discussion

For an overview of the quasi-periodic Hopf bifurcation, see[35,42,45,68,69,123].The quasi-periodic saddle-node and period doubling bifurcations have a similarstructure [69], although the saddle-node case is more involved. For a Hamiltonianversion of the latter case see [195]. For an early treatment of such torus bifurca-tions with only one parameter, see [102,103].

6.2.1 Fraying

We summarize the above results as follows. The quasi-periodic bifurcations tosome extent are similar to their periodic analogues. However, as already seen inthe Hopf–Neımark–Sacker bifurcation, the phenomenon of normal-internal res-onances (which in this case leads to Arnold resonance tongues) is only visiblewhen an extra parameter is taken into account. For quasi-periodic bifurcationseven more parameters may be needed.

In fact, the main difference from the periodic theory is thatthe presence ofresonances leads to Cantorization, compare with Figure 4. To be more precise,in the periodic theory, the subsets of the parameter space corresponding to non-hyperbolicity are piecewise smooth manifolds. The same holds true also in thetorus case when we would consider only integrable systems. In the nearly inte-grable case, however, the dense set of resonances really interrupts these bifurca-tion boundaries.

In the present dissipative setting, the domains of hyperbolicity can be fattenedto open subsets of the parameter space, that near the normal-internal resonancesleave over strands of bubbles. The total effect of this is called fraying of thebifurcation boundaries.

6.2.2 Non-parallel dynamics

The parameter domains with quasi-periodic tori are nowheredense and of posi-tive measure. In the open domains with normally hyperbolic tori, several types ofdynamics can occur: already in3-tori, next to quasi-periodicity one meets period-icity (phase lock) and chaos [326].

Inside a bubble we are closer to a resonance of the form〈ω, k〉 + β` = 0.For ` = 0 this is an internal resonance, while for` 6= 0 the resonance is normal-internal. Compare with [19,71,100,101,176,225,266–268,284,376,404,417,430]for research in this direction; some of these works are case studies, while othershave a more generic point of view. For a Hamiltonian analoguesee [56, 57]. Oneimportant aspect is the repetition of the whole scenario within bubbles. The near-resonance dynamics is quite rich, and may involve cantori and chaos.

57

6.2.3 Final remarks

The quasi-periodic bifurcation theory sketched above has been extended and ap-plied in various directions. A direct generalization of thequasi-periodic saddle-node bifurcation to the case of cusps and higher order degenerate bifurcations isgiven in [432].

The quasi-periodic response problem. A widely used context for applicationsof KAM Theory is that of response solutions in quasi-periodicallyforced oscil-lators. Again, here we consider only the dissipative case, which goes back toJ.J. Stoker [410]. To be definite, we identify the leading example, where a freeDuffing – Van der Pol oscillator is forced as follows:

u + (a + cu2)u + bu + du3 = εf(ω1t, . . . , ωmt, u, u, a, b, ε),

also compare with§ 3.1.3. Here the perturbationf is assumed to be2π-periodicin each of its firstm arguments. This non-autonomous second order differentialequation can be written as a vector field (in the system form)

xj = ωj, j = 1, . . . , m,

u = v, (38)

v = −(a + cu2)v − bu − du3 + εf(x1, . . . , xm, u, v, a, b, ε),

defined on the phase spaceTm × R2 =(x1, . . . , xm; u, v)

. We consider

(a, b) ∈ R2 (varying over an open domain) as a multi-parameter. In this settingthe frequency vectorω ∈ Rm is fixed, with rationally independent components,which is why the forcing is said to be quasi-periodic. The functionf is assumed tobe of classC∞. Finally ε ∈ R, with |ε| 1, is as usual a perturbation parameter.

The response problem asks for the existence of quasi-periodic solutions withthe fixed frequency vectorω. This problem reduces to that of an invariantm-torusof (38) in the phase spaceTm×R2, which is a graphy = y(x) overTm×0; eachsuchm-torus so corresponds to anm-parameter family of response solutionsy =y(x(t)

). Here we denotey = (u, v). In the case of strong damping|a| 1 (with a

either positive or negative) the problem is solved in [410]:in a more contemporarylanguage, it reduces to the persistence of normally hyperbolic invariant m-toriclose toTm×0 ⊂ Tm×R2, compare with§ 4.2. In the case of small damping|a|a quasi-periodic Hopf bifurcation occurs, for details see [35] and for a discussion[42, 68, 69, 123]. The dynamics associated to normal-internal resonance bubblesis treated in [71,176,376,430].

REMARK . The response problem has many analogues in the Hamiltonianandreversible contexts, compare with e.g. [64,73,220,230,310,311,333,401,426].

58

The skew Hopf bifurcation. An extension of the above theory is formed by theskewHopf bifurcation, in its simplest form taking place as a diffeomorphism ofthe solid torusT1 × C:

Pβ,c : T1 × C → T1 × C;

(x, z) 7→(x + 2πβ, czeikx

)+ higher order terms, (39)

whereβ and c are real parameters and we takec > 0. Moreover,k ∈ Z is adiscrete ‘parameter’. The system (39) turns out to be non-reducible wheneverk 6= 0.

Integrability in this case amounts to rotational symmetry in thez-direction. Inthe integrable case we see that the circleTβ,c = T1 × 0 is invariant, being ahyperbolic attractor for0 < c < 1 and a hyperbolic repeller forc > 1. Moreover,it turns out that forc > 1, a 2-torus attractor or repellerT ′ is born, where nearc = 1 resonance problems occur, leading to both Cantorization and fraying inthe(β, c)-plane. The corresponding perturbation problem was studied both in theintegrable [82, 85] and in the nearly integrable [85, 88, 417, 429] cases. As saidbefore in§ 5.2.3, this was an early, successful attempt to developKAM Theory fornon-reducible systems.

REMARK . In the integrable case, system (39) turns out to have mixed powerspectrum, which may have some interest for certain experiments with rotationalsymmetry, where a mixed spectrum occurs [82].

Onset of turbulence. The quasi-periodic Hopf bifurcation has an interest forthe onset of turbulence as described by the theories of Landau–Hopf–Lifshitz andof Ruelle–Takens [217, 252, 253, 361–363]. The idea is to view this aspect offluid dynamics in finitely many dimensions, the number of which is increasingwhen the turbulence gets more developed. For instance, a stationary fluid flowcorresponds to equilibrium dynamics. In the initial Landau–Hopf–Lifshitz theory[217, 252, 253], there are repeated Hopf bifurcations, where the dynamics staysquasi-periodic, but picks up more and more frequencies, thereby complicating thephase portrait. Later the Ruelle–Takens theory [361–363] modified this pictureby pointing out that already in3-tori there can be strange attractors with chaoticdynamics.

We observe that both scenarios have been unified in the present generic theory.For another example how to incorporate quasi-periodicity with n frequencies (forn ∈ N arbitrary) in an infinite dimensional dynamical system, see[45, 164]. Forfurther discussion also see [68,69,123].

59

ω2/ω1

q

Figure 6: Sketch of the Cantorized bifurcation set of the quasi-periodic center-saddle bifurcation forn = 2 [195, 196, 199], where the horizontal axis indicatesthe frequency ratioω2 : ω1. The lower part of the figure corresponds to hyperbolictori and the upper part to elliptic ones.

7 Quasi-periodic bifurcation theory in other settings

The marriage ofKAM Theory with Bifurcation Theory extends far beyond thedissipative setting, but largely follows the same approach.

7.1 Hamiltonian cases

As an example consider the Hamiltonian center-saddle bifurcation, a SingularityTheory model of which is given by

H(p, q) = 12p2 + Vµ(q), whereVµ(q) = 1

3q3 − µq,

with the phase spaceR2 = (p, q) and with one real parameterµ, comparewith [13,199,418]. The corresponding system reads

q = p

p = −dVµ

dq(q).

(40)

The equilibria are given by the equations

p = 0, q2 − µ = 0, (41)

which determines a curve in the product(p, q, µ) of the phase space and theparameter space. This curve is smoothly parametrized byq. The equilibria are

60

elliptic for q > 0 and hyperbolic forq < 0. For µ = 0, a parabolic singularity(fold) occurs atq = 0. The problem is what happens to this scenario when (40) ischanged to the integrable Hamiltonian system

x = ω(y, p, q)

y = 0

q = p

p = −dVµ

dq(q),

(42)

with x ∈ Tn andy ∈ Rn, and even more, what happens when nearly integrableHamiltonian perturbations of (42) are considered. As in theprevious section, onedirectly sees that in the integrable case the (relative) equilibria of (40) correspondto invariantn-tori of (42), which can be either elliptic or hyperbolic. Incorporatingnearly integrable perturbations again requiresKAM Theory [195].

It turns out that the bifurcation set (41) becomes Cantorized by the resonancesand the corresponding Diophantine conditions

∣∣〈ω(y, p, q), k〉∣∣ > γ|k|−τ for q < 0 and∣∣〈ω(y, p, q), k〉+ β`∣∣ > γ|k|−τ for q > 0,

as usual, for allk ∈ Zn \ 0 and for all ` ∈ Z with |`| 6 2, which appar-ently differ for the hyperbolic case (q < 0) and the elliptic case (q > 0). Hereβ =

√2√

µ =√

2q is the normal frequency in the elliptic case, as before be-ing the positive imaginary part of the purely imaginary eigenvalueiβ associatedto the normal linear part. For a sketch of the bifurcation set, see Figure 6. Infact, as before, there is a Whitney smooth conjugation between the integrable andnearly integrable tori, slightly deforming the nowhere dense domains of Diophan-tine quasi-periodic tori. We observe that, in the Hamiltonian setting, even in thehyperbolic case,no fattening occurs. This is due to the equationy = 0 in (42):the ‘hyperbolic’ invariant tori in Hamiltonian systems arenot ‘normally hyper-bolic’ in the sense of the Center Manifold Theory [117,162,215,424] (see§§ 8.2and 8.3.1 below, as well as a detailed discussion in [27]).

The phase space variables(y, q) can ‘absorb’ the external parameters(ω, µ),under suitable nondegeneracy conditions of Kolmogorov (including the iso-energeticcounterpart) or Russmann types, as described in§§ 4.3.2 and 5.2.2.

The study of quasi-periodic bifurcations in the Hamiltonian setting has beendeveloped further for normally parabolic and normally umbilic torus bifurca-tions related to simple singularities and to singularitiesinvolving model param-eters [59, 60, 195–199]. We emphasize that these results do not include a strictapplication of the structure preserving ParametrizedKAM Theory as mentioned

61

above in§ 5.2.1 and described in detail below (for the Hamiltonian case) in§ 8.3.1,since the analogue of the conditiondet Ω(µ0) 6= 0 is violated. The bifurcationsets of the integrable approximations exhibit the familiarhierarchy in the stratifi-cations of Catastrophe Theory [11,13,15,16,63,87,180,286,418], and the nearlyintegrable perturbations yield a Cantorized version of these bifurcation sets.

Another branch of this theory studies the quasi-periodic Hamiltonian Hopfbifurcation [54, 55, 216], built on the normal1 : −1 resonance and illustratedby the Lagrange top [132] subject to periodic and quasi-periodic forcing. HereParametrizedKAM Theory can be directly used. Within this bifurcation, higherdimensional tori branch off, for an overview see [199]. For acase study of theHamiltonian normal1 : 1 : . . . : 1 resonance, see [228]. For a case study ofdynamical effects of Hamiltonian normal-internal resonances, see [56,57], whichis a conservative counterpart of [71,376].

7.2 Discussion

We already mentioned reversibleKAM Theory several times in§§ 5.2.1 and 5.2.2,so now we just refer to [48,49,121,122] for a study in the normal 1 : 1 resonance,that generically involves the quasi-periodic reversible Hopf bifurcation.10

We summarize that normal-internal resonances give rise to Cantorization ofthe corresponding ‘classical’ bifurcation diagrams in thecase of (relative) equi-libria or periodic solutions. These ‘classical’ diagrams,up to diffeomorphisms,are real semi-algebraic sets obtained from Singularity Theory. The Cantorizedobjects sometimes have been calledCantor stratifications. Moreover, in certainhyperbolic cases, the nowhere dense domain of Diophantine quasi-periodic tori asobtained so far, can be fattened. We however emphasize that this yields invarianttori that donot have to be quasi-periodic.

Quasi-periodic bifurcation theory becomes important in higher dimensionalmodelling when dense sets of resonances can lead to Cantorization of bifurcationsets. For examples see [45,80,99–101,225,284,425].

8 Further Hamiltonian KAM Theory

In this section we discussKAM Theory for invariant tori in Hamiltonian systems,other than the Lagrangean case, which is familiar from§ 4.3. This concerns lowerdimensional isotropic tori, higher dimensional coisotropic tori, and ‘atropic’ tori(which are neither isotropic nor coisotropic). We treat thecase of lower dimen-sional isotropic tori in great detail and consider also the possible excitation of

10In the reversible setting, there is no difference between the1 : 1 and the1 : −1 resonance.

62

elliptic normal modes of these tori. However, we start with certain properties ofthe Lagrangean tori and describe their ‘exponential condensation’ and ‘superex-ponential stickiness’ as well as the mechanisms of the destruction of resonant tori.

8.1 Exponential condensation

In §§ 8.1 and 8.2, our concern is some addenda to the LagrangeanKAM Theorem 6,and we retain all the notations and concepts of§§ 4.3 and 4.4 (see especially§ 4.3.1). Consider again a Hamiltonian vector fieldX (with n > 2 degrees offreedom) close to an integrable Hamiltonian fieldX = XH defined onTn ×P , P ⊆ Rn being open and connected. Let us assumeX andX to satisfy thefollowing.

1. Both theX andX are real analytic.

2. X is Kolmogorov nondegenerate onTn × A for an open subsetA ⊆ P .

3. The HamiltonianH is quasi-convexonA. This means that⟨(Dyω)η, η

⟩6=

0 whenevery ∈ A, η ∈ Rn \ 0, and 〈η, ω(y)〉 = 0; recall thatDyωis the derivative of the frequency mapω : P → Rn, ω = ∂H/∂y, and(x ∈ Tn, y ∈ P ) are the angle-action coordinates forX.

The concept of quasi-convex integrable Hamiltonians was introduced by N.N. Nekhoro-shev [323]. Quasi-convexity ofH means strict convexity of the correspondingunperturbed energy hypersurfacesH−1(c) regarded as surfaces inA (rather thanin Tn × A). The properties of Kolmogorov nondegeneracy and quasi-convexityare independent (as simple examples show, see [68],§ 4.2.3) but quasi-convexityimplies iso-energetic nondegeneracy [278] (see§ 4.3.2) and is equivalent to iso-energetic nondegeneracy forn = 2 [278,323].

The systemX is Kolmogorov nondegenerate. Consequently, according to The-orem 6, if X lies in a sufficiently small neighborhoodO of X (say, in theC∞-topology), thenX possesses a family of Diophantine quasi-periodic Lagrangeaninvariantn-tori (KAM tori) in Tn × A. These tori are close to the unperturbedtori Tn × y, and their unionW fills up positive measure. Actually, sinceX isalso iso-energetically nondegenerate, the union ofX-invariant tori fills up positivemeasure even on each perturbed energy hypersurface in the phase space.

THEOREM 15 (Exponential condensation and superexponential stickiness ofKAM tori) [307].Under the three conditions above, letX be fixed and sufficiently close toX andlet T be an arbitrary fixedKAM torus ofX. Denote byUρ(T ) theρ-neighborhoodof T in the phase space. Then the measure ofUρ(T ) \ W is at most of the orderof exp(−c∗/ρ) as ρ ↓ 0, wherec∗ > 0 is a certain constant. Moreover, if the

63

frequency vector of torusT is (τ, γ)-Diophantine then all theX-evolution curvesstarting at a distanceρ < ρ∗ from T stay nearT over an exceedingly long timeThold of the order of

exp

exp

[(ρ∗

ρ

)1/(τ+1)]

,

whereρ∗ > 0 is again a certain constant.

One says that theKAM tori ‘exponentially condense’ to each of them [393]and that each torus is ‘superexponentially sticky’. The latter property was alreadymentioned at the end of§ 4.4.2.

REMARKS.

1. We like to note that the phenomena described here are related to the follow-ing. We consider the completely general set-up of Theorems 5and 12 andthe ensuing remarks. In all these casesKAM tori are density points of quasi-periodicity [68, 69]. If the ambient setting is of classCk, for k ∈ N ∪ ∞large, the corresponding estimates will be only polynomial. In the real an-alytic category, these estimates can be directly sharpenedto exponential orsuperexponential [75,76,320,406].

2. Any Cantor setC is perfect, which means that each of its pointsa is anaccumulation point of the complementC \ a. Every torus among theKAM tori that ‘condense’ toT in Theorem 15 is, in turn, a ‘condensationpoint’ of otherKAM tori, and so on. This hierarchy (which seems to be notcompletely understood yet) is described and discussed in [185,306,308].11

3. The ‘exponential condensation’ ofKAM tori proven in [307] was confirmednumerically (for a model problem of area preserving maps of the plane, i.e.,in the context of Moser’s Twist Mapping Theorem 3) in [166,258], see alsoa discussion in [165].

4. Most probably, the quasi-convexity condition imposed onthe unperturbedHamiltonianH in Theorem 15 is a purely technical one and can be omit-ted. There is also no doubt that Theorem 15 can be carried overto Gevreysmooth systems (perhaps with a somewhat worse estimate of the timeThold).To the best of the authors’ knowledge, these improvements have not beenproven yet. Note, however, that the closely related question of the valid-ity of Nekhoroshev estimates (see the end of§ 4.4.2) for Gevrey smoothsystems has been solved affirmatively in [294,377].

11Similar open questions exist in the characterization of theDiophantine setsDdτ,γ(Rn) ⊆

Dτ,γ(Rn) as mentioned in§ 5.1.1.

64

8.2 Destruction of resonant tori

Here we again consider a small Hamiltonian perturbationX of an integrableHamiltonian systemX = XH (with n > 2 degrees of freedom) assumed to beKolmogorov nondegenerate onTn × A, A ⊆ Rn. According to Theorem 6, Dio-phantine quasi-periodic invariantn-tori Tn × y∗ of X (y∗ ∈ A) give rise to La-grangean invariantn-tori of X (KAM tori) with the same frequency vectorsω(y∗),whereω = ∂H/∂y. In between theKAM tori, there lie the so-calledresonantzonesof X. What is the fate of an unperturbed torusTn × y∗ in the oppositecase, where the frequenciesω1(y

∗), ω2(y∗), . . . , ωn(y

∗) are rationally dependent(so that the torus in question is resonant and is foliated into invariant tori of asmaller dimension)? To fix thoughts, suppose that among then components ofthe vectorω(y∗) = ω∗, there aren− l (1 6 l 6 n− 1) strongly incommensurablenumbers

ω∗i1, ω∗

i2, . . . , ω∗

in−l, (43)

whereas the remainingl componentsω∗j1, ω

∗j2, . . . , ω

∗jl

are rational combinationsof the numbers (43)—so that the frequencies of the torusTn × y∗ satisfy lindependent resonance relations. Then the following statement holds.

THEOREM 16 (Destruction of resonant unperturbed tori).Let X be Kolmogorovnondegenerate. Then for generic sufficiently small perturbations X of X, theinvariant n-torusTn × y∗ of X with the properties indicated above breaks upinto a finite collection of Diophantine quasi-periodic invariant (n − l)-tori of Xwhich lie in a resonant zone.

We will not try to attach any exact meaning to the word ‘generic’ here (variousprecise versions of Theorem 16 can be found in [107, 124, 261,262, 277, 419,435]). In fact, in the resonant zones, as a rule, all the complexity of SingularityTheory occurs. This leads to a Cantor stratification in the neighborhood of anysingle resonance, compare with [61]. Let us note the following, regarding thecodimension0 strata where the tori are elliptic or hyperbolic or of mixed type,compare with a discussion in [68,399]. Under some mild conditions, via a certainrather standard averaging and truncation procedure described in detail in e.g. [17],one can reduce the perturbed Hamiltonian of the systemX near the torusTn ×y∗ to the form

H(χ, J, K) =⟨ω∗, K

⟩+ H(χ, J), H(χ, J) = 1

2〈BJ, J〉 + V (χ),

whereω∗ ∈ Rn−l is the vector with components (43),χ ∈ Tl, J ∈ Rl, K ∈Rn−l, B is a real symmetricl × l matrix (det B 6= 0), and |V | 1. It turnsout that generically to each nondegenerate critical pointχ? of the functionV ,there ‘corresponds’ (in the sense to be made precise) a Diophantine quasi-periodic

65

invariant(n − l)-torus of the systemX. This torus lies in a small vicinity of theoriginaln-torusTn × y∗. The word ‘nondegenerate’ here means that

det∂2V (χ?)

∂χ26= 0.

On the other hand,χ? ∈ Tl is a nondegenerate critical point of the potentialVif and only if the point(χ?, 0) is a nondegenerate equilibrium of the Hamiltoniansystem withl degrees of freedom and the HamiltonianH (the symplectic formbeingdχ ∧ dJ). In the latter case, the word ‘nondegenerate’ means that all theeigenvalues of this equilibrium, i.e. the eigenvalues of the2l × 2l matrix

(0 B

−∂2V (χ?)/∂χ2 0

), (44)

are other than zero.A nondegenerate equilibrium of a Hamiltonian system can beelliptic (all the

eigenvalues are purely imaginary),hyperbolic(all the eigenvalues lie outside theimaginary axis), andof mixed type. According to the type of equilibria(χ?, 0) ofthe system with HamiltonianH, the corresponding invariant(n − l)-tori of thesystemX are also said to beelliptic, hyperbolic, andof mixed type(see [27] fora detailed discussion on the relations between this definition of hyperbolicity ofinvariant tori in Hamiltonian flows and the general concept of hyperbolic invariantmanifolds in dynamical systems as presented in e.g. [117,162,215,424]). The caseof hyperbolic(n− l)-tori in Theorem 16 turns out to be much easier than the caseof non-hyperbolic tori (i.e., elliptic tori and tori of mixed type).

For instance, suppose that the Hamiltonian ofX has the formH(y)+εh(x, y, ε).Then the potentialV has the formV (χ) = εv(χ, ε), and for every sufficientlysmallε > 0, the systemX possesses a hyperbolic invariant(n − l)-torus ‘emerg-ing’ from a given hyperbolic equilibrium(χ?, 0) of the system withl degrees offreedom and the Hamiltonian

limε↓0

ε−1H(χ, ε1/2J

)= 1

2〈BJ, J〉 + v(χ, 0) (45)

(provided that the functionh(·, ·, 0) is generic). In the analytic category this torusdepends onε analytically [419]. A non-hyperbolic equilibrium(χ?, 0) of the sys-tem with the Hamiltonian (45) gives rise to a non-hyperbolicinvariant(n−l)-torusof the systemX only for the most values (in the sense of Lebesgue measure) ofthe perturbation parameterε.

Historical remarks. The casel = n−1 of ‘maximal’ resonance (where the toriin question are in fact circles) in Theorem 16 was consideredalready by Poincare

66

(for a modern presentation see, e.g., [68]). This classicalresult lies outsideKAM

Theory because it does not involve small divisors. The case of arbitrary l wasexamined only in 1989 (that is 35 years afterKAM Theory was founded in Kol-mogorov’s paper [238]!) by D.V. Treshchev [419]. But Treshchev treated onlyhyperbolic invariant(n − l)-tori. The hyperbolicl = 1 case was also explored in-dependently in subsequent papers [104, 152, 331, 359, 423, 434]. Non-hyperbolicinvariant tori in Theorem 16 were first constructed in [105] for the casel = 1(note that ifl = 1 then the tori are either hyperbolic or elliptic). The general caseof Theorem 16—an arbitraryl and arbitrary type of the invariant(n − l)-tori—was announced by Ch.-Q. Cheng and Sh. Wang [107, 435]. In fact, Cheng andWang [107, 435] considered only the case where the eigenvalues of the matrix(44) are either real or purely imaginary (quadruplets±a ± ib of complex eigen-values were excluded). Finally, F. Cong, T. Kupper, Y. Li, and J. You [124] andindependently Y. Li and Y. Yi [261,262] proved Theorem 16 forarbitraryl and ar-bitrary type of the invariant(n− l)-tori (and arbitrary collections of eigenvalues).Some degenerate cases were examined in [192]. Another approach to constructinginvariant(n − l)-tori with arbitraryl and arbitrary normal behavior in the contextof Theorem 16 was proposed in [277]. In [173,174,177,178], elliptic, hyperbolic,and mixed type invariant tori in Theorem 16 were obtained by anew method (thepaper [174] is devoted to the casel = 1 in the presence of some degeneracies).Thus, now we possess a complete picture of the destruction ofresonant tori ofintegrable Hamiltonian systems under small perturbations.

The papers [104, 105, 124, 152, 173, 174, 177,178,192, 261, 262, 331, 359, 419,423,434] cited above studied the analytic situation whereas the articles [107,277,435] dealt with finitely smooth systems. In works [152, 277, 331, 359, 419, 423],a special attention was paid to then-dimensional separatrix stable and unsta-ble manifolds (‘whiskers’) of the hyperbolic invariant(n − l)-tori one looks for.Such ‘whiskers’ are of great importance in the Arnold diffusion mechanism (see§ 4.4.2).

Theorem 16 admits reversible analogues [269, 437]. The papers [269, 437]consider an arbitrary number of resonance relations and arbitrary type of the tori.Some analogues of Theorem 16 for symplectic diffeomorphisms are presentedin [229].

8.3 Lower dimensional isotropic invariant tori

Our discussion ofKAM Theory for lower dimensional isotropic invariant tori inHamiltonian systems is based on the parametrized approach as expounded in de-tail for the dissipative case in§ 5.2. Here the words ‘lower dimensional’ meanthat the dimensions of the tori are smaller than the number ofdegrees of freedomand the word ‘isotropic’ means that the restriction of the symplectic form to the

67

tori in question vanishes.We start with the Hamiltonian analogue of Theorem 12. In Remark 2 after the

formulation of this theorem in§ 5.2.1 we already pointed out that Theorem 12 hasanalogues for various structure preserving settings. Nextwe explain how one canget rid of external parameters and obtain aKAM theorem for lower dimensionalinvariant tori in individual Hamiltonian systems satisfying Russmann-like nonde-generacy conditions via the so-called Herman method already mentioned at theend of§ 5.2.2. Historical remarks will conclude the topic.

8.3.1 The parametrized HamiltonianKAM theorem

TakeM = Tn × Pact × R2m = (x, y, z) as the phase space endowed by thesymplectic form

σ =n∑

j=1

dxj ∧ dyj +m∑

j=1

dzj+m ∧ dzj ,

wherePact ⊆ Rn = y is an open and connected subset. Let alsoPpar ⊆ Rs =µ, an open subset, be the parameter space. Here the subscripts‘act’ and ‘par’are for ‘action’ and ‘parameter’, respectively. OnM , consider aC∞-family ofHamiltonians

H = Hµ(x, y, z) = E(y, µ) + 12〈B(y, µ)z, z〉 + h(x, y, z, µ), (46)

whereE : Pact × Ppar → R and B : Pact × Ppar → gl(2m, R) are certainC∞-mappings, whileh = O(|z|3) with the O-estimate uniform iny andµ (theuniformity in x is automatic due to the compactness ofTn). The2m× 2m matrixB(y, µ) is supposed to be symmetric for all the values ofy andµ. The corre-sponding family of vector fieldsX = XHµ

(x, y, z) is given by

x = ∂Hµ/∂y = ω(y, µ) + O(|z|2)y = −∂Hµ/∂x = O(|z|3) (47)

z = J (∂Hµ/∂z) = Ω(y, µ)z + O(|z|2),

compare with (25), where

ω(y, µ) = ∂E(y, µ)/∂y, Ω(y, µ) = JB(y, µ), (48)

andJ ∈ GL(2m, R) denotes the ‘symplectic unit’:(

m × m zero matrix −(m × m identity matrix)m × m identity matrix m × m zero matrix

).

68

Note thatJ T = J −1 = −J where the superscriptT means transposing.The vector fields (47) are not integrable in the sense of§§ 4.2.1, 4.3.1, and 5.2.1

because we allow the remainderh to bex-dependent, but this is irrelevant for ourpurposes since the normal linear parts of (47) given by

ω(y, µ)∂

∂x+ Ω(y, µ)z

∂

∂z, (49)

compare with (26), are integrable. In particular, on the invariant2n-dimensionalsurfaceΠ = z = 0 = Tn × Pact × 0 we obtain Liouville integrable Hamil-tonian dynamics (see§ 4.3.1). Indeed, then-tori Tyµ = Tn × y × 0 areXHµ

-invariant for allµ and the restriction ofXHµto these tori is conditionally pe-

riodic (see§ 4.1), the corresponding frequency mapω : Pact × Ppar → Rn beingdefined by (48). Note that the toriTyµ are Lagrangean withinΠ and are isotropicwhen considered as submanifolds of the whole phase spaceM .

As usual, we are interested in the fate of the toriTn×y×0, y ∈ Pact, undersmall HamiltonianC∞-perturbationsX of X, for the ‘action’ variabley rangingnear a certain pointy0 ∈ Pact and for the parameter valuesµ near a certain pointµ0 ∈ Ppar. In the sequel, we mimic the exposition of§ 5.2.1.

BHT nondegeneracy. As in the dissipative case treated in§ 5.2.1, the suitableBHT nondegeneracy condition for the unperturbed vector fields (47) in the presentHamiltonian set-up involves the frequenciesω1(y, µ), . . . , ωn(y, µ) of the toriTyµ

(the internal frequencies) as well as the eigenvalues of thecorresponding ‘normal’matricesΩ(y, µ). The latter matrices are Hamiltonian (= infinitesimally symplec-tic) in the sense thatΩJ + JΩT = 0 for eachy andµ:

ΩJ + JΩT = JBJ + JBTJ T = JBJ − JBJ = 0.

Thus,ΩT = JΩJ = −JΩJ −1, whence−Ω has the same spectrum and the sameJordan structure asΩT and, consequently, asΩ. In particular, the eigenvalues ofΩ come in pairs±λ.

As in § 5.2.1, we will confine ourselves to the case where all these eigenvaluesare simple for(y, µ) near(y0, µ0), then automaticallydet Ω(y, µ) 6= 0 for such(y, µ). Suppose that the eigenvalues ofΩ are given by

(±δ1, . . . ,±δN1 , ±iζ1, . . . ,±iζN2 , ±α1 ± iβ1, . . . ,±αN3 ± iβN3

)(50)

with positiveδj , ζj , αj , βj, compare with (27). Note thatN1 + N2 + 2N3 = m.We call

(ζ, β) =(ζ1, . . . , ζN2, β1, . . . , βN3

)

thenormal frequenciesof the invariant torus. The map

spec : Ω 7→ (δ, ζ, α, β) ∈ RN1 × RN2 × RN3 × RN3 ,

69

compare with (28), parametrizes nearΩ(y0, µ0) the orbit space of the group of2m× 2m symplectic matrices (= the group of2m× 2m matricesS subject to theequalitySJST = J ) acting on the set of2m × 2m Hamiltonian matrices. TheBHT nondegeneracy condition in this case consists in that the map

Pact × Ppar 3 (y, µ) 7→(ω × (spec Ω)

)(y, µ) ∈ Rn × RN1 × RN2 × RN3 × RN3 , (51)

compare with (29), is a submersion in an open neighborhoodA ⊆ (Pact×Ppar) of(y0, µ0) [67–69,223]. According to the Inverse Function Theorem [214,318,407],this condition is equivalent to that the derivative of the map (51) is surjective aty = y0, µ = µ0. We then say that the familyX = XHµ

is nondegenerate onthe torus unionTn × 0 × A =

⋃(y,µ)∈A Tyµ, recall that here0 ∈ R2m. Unlike

to § 5.2.1, we will not try to suppress extra parameters that may possibly occurbecause such a suppression could a priori affect not onlyµ but the ‘action’ variabley as well. Without loss of generality the domainA can be assumed to have theform A = Aact × Apar with Aact ⊆ Pact andApar ⊆ Ppar.

Diophantine conditions. To formulate appropriate Diophantine conditions onthe internal and normal frequencies, forτ > n − 1 andγ > 0 we define the set of(τ, γ)-Diophantine normal-internal frequency vectors by

Dτ,γ(Rn; RN2+N3) =

(ω, ζ, β) ∈ Rn × RN2 × RN3

∣∣ |〈ω, k〉 + 〈ζ, ł〉 + 〈β, `〉| > γ|k|−τ , (52)

for all k ∈ Zn \ 0 and for allł ∈ ZN2 , ` ∈ ZN3 with |ł | + |`| 6 2,

compare with (30). This is again a nowhere dense set of positive measure (forγ sufficiently small) that possesses the closed half line property, see§§ 4.1.2and 5.1.1; compare with [67–69,223,311,312]. LetΓ =

(ω × (spec Ω)

)(A), as

in § 5.2.1,

Γγ =(ω, δ, ζ, α, β) ∈ Γ

∣∣ dist((ω, δ, ζ, α, β), ∂Γ

)> γ

,

andDτ,γ(Γγ) = Γγ

⋂(Dτ,γ(R

n; RN2+N3) × RN1δ × RN3α).

The further definition ofDτ,γ(Aγ) ⊂ A is now obvious, compare with§§ 4.2.1and 4.3.1. Again, note that the closed half lines ofDτ,γ(R

n; RN2+N3) now turninto closed linear half spaces of dimension1+N1 +N3 and that these geometricalstructures, up to a diffeomorphism, are inherited by the perturbations.

70

THEOREM 17 (ParametrizedKAM – Hamiltonian isotropic case) [62,69,216,223].Letn > 2. Let theC∞-familyX = XHµ

(x, y, z) of vector fields(47)beBHT non-degenerate onTn×0×A, withA = Aact×Apar ⊆ (Pact×Ppar) open. Then, forγ > 0 sufficiently small, there exists a neighborhoodO of X in theC∞-topology,such that for any perturbed family of Hamiltonian fieldsX ∈ O there exists aC∞-mapping

Φ : Tn × Aact × R2m × Apar × Aact → Tn × Aact × R2m × Apar,

defined nearTn × Aact × 0 × Apar × Aact, with the following properties:

1. For eachy′ ∈ Aact, the mapping

Φ[y′] : Tn × Aact × R2m × Apar → Tn × Aact × R2m × Apar

defined asΦ[y′](x, y, z, µ) = Φ(x, y, z, µ, y′)

is aC∞-near the identity map preserving projections to the parameter spacePpar.

2. For eachy′ ∈ Aact, the image of theX-invariant torus union

Tn × y′ × 0 ×µ ∈ Apar

∣∣ (y′, µ) ∈ Dτ,γ(Aγ)

underΦ[y′] is X-invariant, and the restricted map

Φ[y′] = Φ[y′] |Tn×y′×0×µ∈Apar | (y′,µ)∈Dτ,γ(Aγ)

conjugatesX to X, that is

Φ[y′]∗X = X.

3. Moreover,Φ[y′] preserves the normal linear behavior of the toriTy′µ for(y′, µ) ∈ Dτ,γ(Aγ) with respect toX (see item 3 of Theorem 12 for anexplanation).

REMARKS.

1. Roughly speaking, Theorem 17 relates to Theorem 12 of§ 5.2.1 as the La-grangeanKAM Theorem 6 of§ 4.3.1 does to the normally hyperbolicKAM

Theorem 5 of§ 4.2.1 (and as the twist Theorem 3 of§ 3.2.1 does to the

71

circle-map Theorem 2 of§ 3.1.2). The conclusion of Theorem 17 first of allexpresses that the familyX (47) is quasi-periodically stable on the union

⋃

(y,µ)∈Dτ,γ (Aγ)

Tyµ = Tn × 0 × Dτ,γ(Aγ)

of Diophantine quasi-periodic invariantn-tori [67–69, 223], compare with§§ 3.1.3, 4.2.1, 4.3.1, 5.2.1 above. Taking item 3 of Theorem 17into ac-count, we also speak ofnormal linear stabilityof this torus union, comparewith Remark 1 after Theorem 12.

2. Introducing the additional parametery′ ∈ Aact is called alocalization pro-cedure. This trick is explained in [62,69,216,223] in detail, see also the be-ginning of§ 5.2.2. Note that for some particular values ofy′, the Lebesguemeasuremeas [y′] of the set

µ ∈ Apar

∣∣ (y′, µ) ∈ Dτ,γ(Aγ)

can be zeroand this set can even be empty, however smallγ is. Nevertheless,∫

Aact

meas [y′] dy′

tends to the measure ofA asγ ↓ 0 according to the Fubini theorem.

3. The mapsΦ[y′] (to be more precise, their phase space components) are ingeneral not symplectic (compare with Theorem 6).

4. As we already indicated in Remark 3 after Theorem 12, the assumption ofa simple spectrum ofΩ(y0, µ0) can be dropped, the key references for thisin the present Hamiltonian setting being [62,216].

The unperturbedX-invariant toriTyµ, (y, µ) ∈ A, as well as the perturbedX-invariant toriΦ[y](Tyµ), (y, µ) ∈ Dτ,γ(Aγ), are said to be

1. elliptic, if all the eigenvalues ofΩ(y, µ) are purely imaginary (i.e.N1 =N3 = 0, N2 = m);

2. hyperbolic, if all the eigenvalues ofΩ(y, µ) lie outside the imaginary axis(i.e.N2 = 0, N1 + 2N3 = m);

3. of mixed type, if some eigenvalues ofΩ(y, µ) are purely imaginary and someare not (i.e.N2 > 0 andN1 + N3 > 0),

compare with§ 8.2. The relation between this definition of hyperbolicity of lowerdimensional invariant tori in Hamiltonian flows and the general concept of hy-perbolic invariant manifolds in dynamical systems (as presented in e.g. [117,162,215, 424]) is discussed in detail in [27]. It is important to emphasize that eachof these three types of normal behavior of invariant tori is an openproperty: if acertain torus is elliptic (hyperbolic, of mixed type), so are all the nearby tori of allthe nearby systems.

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8.3.2 Lower dimensional tori in individual Hamiltonian systems

Quasi-periodic and normal linear stability of invariantn-tori Tyµ in Theorem 17,for (y, µ) ∈ Dτ,γ(Aγ), requires a lot of external parameters: the parameter spacePact should be of dimensions > m to make it possible for the map (51) to bea submersion. Now we will show how one can get rid ofall these parametersand deduce aKAM statement about lower dimensional invariant tori inindividualHamiltonian systems.

We consider the same symplectic phase spaceM = (x, y, z) as in§ 8.3.1(but with the range domain fory denoted byP rather than byPact) and aC∞-HamiltonianH onM of the same form as (46), but without the parameterµ:

H = H(x, y, z) = E(y) + 12〈B(y)z, z〉 + h(x, y, z), (53)

whereE : P → R andB : P → gl(2m, R) are certainC∞-mappings, whileh =O(|z|3) with ay-uniformO-estimate. The2m×2m matrixB(y) is supposed to besymmetric for all the values ofy. The corresponding vector fieldX = XH(x, y, z)is given by

x = ∂H/∂y = ω(y) + O(|z|2)y = −∂H/∂x = O(|z|3) (54)

z = J (∂H/∂z) = Ω(y)z + O(|z|2),

compare with (47), where

ω(y) = ∂E(y)/∂y, Ω(y) = JB(y),

compare with (48). The normal linear part of (54) is

ω(y)∂

∂x+ Ω(y)z

∂

∂z, (55)

compare with (49). Our interest is Diophantine quasi-periodic invariantn-tori ofsmall HamiltonianC∞-perturbationsX of X near the surfaceΠ = z = 0, tobe more precise, near the invariant toriTn × y × 0 of X, y ∈ P .

Diophantine stability. To this end, assume again that fory ranging in an open(and bounded) subsetA ⊆ P , the spectrum of the Hamiltonian matrixΩ is simpleand given by (50) with positiveδj , ζj, αj, βj . Suppose also that the map

ω × ζ × β : P → Rn × RN2 × RN3; y 7→(ω(y), ζ(y), β(y)

)(56)

possesses the following property ofDiophantine stability, related to the theory ofDiophantine approximations on submanifolds of Euclidean spaces, see§ 4.3.2.

73

DEFINITION 18 (Diophantine stability).The map(56) is said to beDiophantinestableif there exists a neighborhoodQ of this map in theC∞-topology such thatfor any sufficiently large positiveτ and for any perturbed map

ω × ζ × β ∈ Q; y 7→(ω(y), ζ(y), β(y)

),

the Lebesgue measure of the sety ∈ A

∣∣ (ω(y), ζ(y), β(y))

/∈ Dτ,γ(Rn; RN2+N3)

tends to0 asγ ↓ 0 uniformly inω×ζ×β ∈ Q. Recall that the setDτ,γ(Rn; RN2+N3)

of (τ, γ)-Diophantine normal-internal frequency vectors is definedin (30) and(52).

The explicit conditions on (56) sufficient for Diophantine stability are known[67, 68, 396] and can be formulated in terms of the partial derivatives ofω, ζ ,andβ of all the orders from0 to some positive integerQ. In particular, theseconditions include Russmann nondegeneracy ofω discussed in§ 4.3.2. We willnot reproduce these conditions here because they are somewhat cumbersome andnot illuminative, but we would like to note the following technical error persistentin many formulas in the works [67,68,396] as well as in the papers [389,392,393,395,398] and in the Russian 2002 edition of the book [17] (dueto an aberration insome calculations of the second author of the present survey): all the expressionsof the form ∑

|q|=J

Dqf(a)bq or∑

|q|=J

〈Dqf(a), g〉bq

in all these works should be replaced with

J !∑

|q|=J

Dqf(a)bq

q!and J !

∑

|q|=J

〈Dqf(a), g〉bq

q!,

respectively. HereJ is a non-negative integer,f : Rν → Rl is a CJ-mapping,a ∈ Rν , b ∈ Rν , g ∈ Rl, all the components of the vectorq ∈ Zν are non-negative, and the standard multi-index notation is used:

Dqf(a) =∂q1+···+qν

∂aq1

1 · · ·∂aqνν

f(a), bq = bq1

1 · · · bqν

ν , q! = q1! · · · qν !.

THEOREM 19 (Hamiltonian lower dimensionalKAM without parameters).Letn >

2. Let the map(56)corresponding to theC∞-vector fieldX = XH(x, y, z) of theform (54) be Diophantine stable on an open and bounded subsetA ⊆ P . Then,for anyε > 0, there exists a neighborhoodO of X in theC∞-topology, such thatfor any perturbed Hamiltonian fieldX ∈ O there exist

74

– a subsetA] ⊂ A,

– a vector field

X]lin(x, y, z) = ω](y)

∂

∂x+ Ω](y)z

∂

∂z, (57)

compare with(55), with C∞-mapsω] : A → Rn andΩ] : A → gl(2m, R),

– and aC∞-mappingΦ : Tn ×A×R2m ×A → Tn ×A×R2m, defined nearTn × A × 0 × A,

that possess the following properties.

1. The Lebesgue measure ofA \ A] is less thanε.

2. The mapsω] and Ω] are C∞-close to the mapsω and Ω, respectively.Moreover, the2m × 2m matrix Ω](y) is Hamiltonian for eachy ∈ A(Ω]J + JΩ]T ≡ 0).

3. For eachy′ ∈ A, the mappingΦ[y′] : Tn × A × R2m → Tn × A × R2m

defined asΦ[y′](x, y, z) = Φ(x, y, z, y′) is aC∞-near the identity map.

4. For eachy′ ∈ A], the image of theX- and X]lin-invariant torusTn ×

y′ × 0 under Φ[y′] is X-invariant, and the restricted mapΦ[y′] =Φ[y′] |Tn×y′×0 conjugatesX]

lin to X, that is

Φ[y′]∗X]lin = X.

5. Finally, for eachy′ ∈ A], Φ[y′] preserves the normal linear behavior of thetorusTn × y′ × 0 with respect toX]

lin.

This theorem is an almost immediate corollary of Theorem 17.

Sketch of the proof of Theorem 19. For y ∈ A, one can easily constructC∞-mappingsE‡ = E‡(y, µ) andB‡ = B‡(y, µ), with µ ranging near the origin0 ofRs (s being sufficiently large), such that

∼ E‡(y, 0) ≡ E(y) andB‡(y, 0) ≡ B(y);

∼ B‡(y, µ) is symmetric for all the values ofy andµ;

∼ theC∞-family of Hamiltonian vector fieldsX‡ = X‡

H‡µ

on Tn × A × R2m

afforded by the Hamiltonians

H‡ = H‡µ(x, y, z) = E‡(y, µ) + 1

2〈B‡(y, µ)z, z〉 + h(x, y, z),

compare with (46) and (53), is BHT nondegenerate on the torusunion⋃µ near0 ∈ R

s Tn × A × 0.

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That the spectrum ofΩ(y) is simple fory ∈ A is sufficient for this, independentlyof whether the map (56) is Diophantine stable or not. Note that H‡

0 ≡ H, X‡

H‡0

=

XH , and for the maps

ω‡(y, µ) = ∂E‡(y, µ)/∂y, Ω‡(y, µ) = JB‡(y, µ),

compare with (48), one hasω‡(y, 0) ≡ ω(y), Ω‡(y, 0) ≡ Ω(y). The eigenvaluesof Ω‡ are given by

(±δ‡1, . . . ,±δ‡N1

, ±iζ‡1 , . . . ,±iζ‡

N2, ±α‡

1 ± iβ‡1, . . . ,±α‡

N3± iβ‡

N3

)

with positiveδ‡j , ζ‡j , α‡

j, β‡j , compare with (50), andδ‡j(y, 0) ≡ δj(y), ζ‡

j (y, 0) ≡ζj(y), α‡

j(y, 0) ≡ αj(y), β‡j (y, 0) ≡ βj(y).

Fix positiveτ sufficiently large and apply Theorem 17 to the families of Hamil-tonian vector fieldsX‡ = X‡

H‡µ

andX‡

H‡µ

+ X −XH . Forγ > 0 sufficiently small

and forX sufficientlyC∞-close toX = XH , we obtain aC∞-mapping

Φ‡ = Φ‡(x, y, z, µ, y′) =(Ψ[µ, y′](x, y, z), Υ[µ, y′]

), (58)

wherex ∈ Tn, y ∈ A, y′ ∈ A, the variablez ranges near0 ∈ R2m, the parameterµ ranges near0 ∈ Rs, andΨ[µ, y′](x, y, z) ∈ Tn × A × R2m, Υ[µ, y′] ∈ Rs. Themapping (58) possesses the following properties:

1. For eachy′ ∈ A, the mapping(x, y, z, µ) 7→ Φ‡(x, y, z, µ, y′) is C∞-nearthe identity (that theµ-componentΥ[µ, y′] of this mapping does not de-pend on the phase space variablesx, y, z just expresses the preservation ofprojections to theµ-space).

2. For eachy′0 ∈ A andµ0 near0 ∈ Rs such thaty′

0 is not too close to theboundary∂A and

(ω‡(y′

0, µ0), ζ‡(y′

0, µ0), β‡(y′

0, µ0))∈ Dτ,γ(R

n; RN2+N3),

then-torus

Ψ[µ0, y′0](Tn × y′

0 × 0)⊂ (Tn × A × R2m) (59)

is invariant under the vector fieldX‡

H‡

Υ[µ0,y′0]

+ X − XH .

3. Moreover, the restriction ofΨ[µ0, y′0] to the torusTn×y′

0×0 conjugatesX‡

H‡µ0

to X‡

H‡

Υ[µ0,y′0]

+ X − XH .

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4. Moreover,Ψ[µ0, y′0] preserves the normal linear behavior of the torusTn ×

y′0 × 0 with respect toX‡

H‡µ0

.

One can solve the equationΥ[µ, y′] = 0 with respect toµ and obtainµ = ξ(y′)where the functionξ : A → Rs is C∞-small. Setµ0 = ξ(y′

0) in the above listof the properties of mapping (58). Forµ0 = ξ(y′

0), the torus (59) isX-invariant.Since the map (56) is Diophantine stable, the measure of the set A‖ of pointsy′ ∈ A such that

(ω‡(y′, ξ(y′)), ζ‡(y′, ξ(y′)), β‡(y′, ξ(y′))

)∈ Dτ,γ(R

n; RN2+N3)

tends to the measure ofA asγ ↓ 0. Now it suffices to set

A] = A‖ \ (a narrow neighborhood of∂A),

ω](y) = ω‡(y, ξ(y)), Ω](y) = Ω‡(y, ξ(y)),

Φ(x, y, z, y′) = Ψ[ξ(y′), y′](x, y, z).

QED, the proof is completed.

The central idea of this proof is artificially introducing parameters into the sys-tem (or adding extra parameters) to achieve BHT nondegeneracy and subsequentlyeliminating these parameters. This idea was proposed by M.R. Herman in his talkat an international conference on dynamical systems in Lyons in 1990, whereHerman gave an exceedingly simple proof of the fact that Russmann nondegen-eracy is sufficient for the presence of many perturbed Lagrangean tori. Hermandid not publish his proof, although his ‘parameter reduction’ method was usedin [449] in a problem outside Hamiltonian mechanics (that problem concernedthe so-called ‘vertically translated’n-tori in Tn × R). In the mid 1990’s, theauthors of the present survey together with G.B. Huitema applied systematicallyHerman’s method (to be treated as a special tool within ParametrizedKAM The-ory) to variousKAM contexts including Lagrangean [67, 68, 390, 391] and lowerdimensional [67,68,396] invariant tori in Hamiltonian systems, as well as invarianttori in reversible [67,68,389], volume preserving, and dissipative [67,68] systems.In [401], Herman’s idea is used to construct invariant tori in quasi-periodic non-autonomous perturbations of dynamical systems preservingvarious structures.In [400], the same method is exploited to study the so-calledpartial preserva-tion of frequencies inKAM Theory explored previously in [119, 264, 271, 272].Some similar approaches were used by J. Fejoz [160] and by A.A. Kubichka andI.O. Parasyuk [242,243].

REMARKS.

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1. The conclusion of Theorem 19 expresses the fact that any Hamiltonian sys-tem X sufficiently close toX admits a Whitney differentiable family ofinvariant n-tori close to the unperturbed toriTn × y × 0 and car-rying quasi-periodic motions. However, there is in generalno connec-tion between the frequencies and the normal behavior of the perturbedX-invariant tori, on the one hand, and the frequencies and the normal behaviorof the unperturbedX-invariant tori, on the other hand (this is typical for alltheKAM theorems with Russmann-like nondegeneracy conditions, comparewith § 4.3.2). The frequency vector and ‘normal’ matrix of every perturbedtorus are equal respectively toω](y) andΩ](y) for somey ∈ A] ⊂ A ratherthan toω(y) andΩ(y). Therefore,in the framework of Theorem 19, wecan speak neither of quasi-periodic stability of the unperturbed tori nor ofnormal linear stability. The unperturbed tori in Theorem 19 do not persistunder small Hamiltonian perturbations, one may just assertthe existence ofmany invariant tori in a perturbed system.

2. Of course, the concept of Diophantine stability and all the setting of Theo-rem 19 can be carried over to the case where the unperturbed system alreadydepends on some external parameters [67,68].

3. Generically, the mappingy 7→ ω](y) = ω‡(y, ξ(y)) in Theorem 19 can-not be represented, even locally, as the frequency map of anyintegrableHamiltonian system withn degrees of freedom: the1-form ω](y) dy =∑n

j=1 ω]j(y) dyj is not closed. Consequently, the vector field (57) is in gen-

eral not Hamiltonian.

4. As in Theorem 17, the mapsΦ[y′] in Theorem 19 are generically not sym-plectic.

5. The assumption of a simple spectrum ofΩ(y) in Theorem 19 can also beomitted, see [263, 441, 454]. There are versions of this theorem wheredet Ω(y) is allowed to vanish [263,453].

8.3.3 Historical remarks

Studies of lower dimensional isotropic invariantn-tori in Hamiltonian systemswere started by V.K. Melnikov [302, 303] and J.K. Moser [311,312] in the mid1960’s. Since then, this topic (in the context where the unperturbed system pos-sesses, for each value of the external parameter if the latter is present, a2n-dimensional surface foliated into unperturbed invariantn-tori) has attracted a wideattention, the most important references probably being [21, 33, 62, 68, 69, 91,115, 125, 127, 150, 187, 192, 216, 223, 230, 231, 263, 264, 271, 347, 370, 426, 441,

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442, 444, 453, 454, 457, 459, 460], see also [17, 27, 54, 67, 111, 112, 114, 160, 199,220, 232, 360, 393, 396, 400, 401, 420, 445, 456]. These worksassume quite dif-ferent nondegeneracy and nonresonance conditions imposedon the mappingsωand Ω. For instance, Russmann-like nondegeneracy conditions for ω are usedin [17, 67, 68, 125, 127, 264, 271, 370, 393, 396, 400, 401, 459, 460]. Resonancesbetween internal and normal frequencies of elliptic tori are considered in the pa-pers [33,442,444,445] (recall that other dynamical settings where normal-internalresonances come in play are examined in e.g. [56, 57, 71, 176,376, 430], see Sec-tions 6 and 7).

As a rule, non-hyperbolic lower dimensional invariant toriin KAM Theory turnout to be much harder to construct than the hyperbolic ones, compare with§ 8.2.For example, first general preservation theorems for hyperbolic lower dimensionaltori were obtained in 1973–74 [21, 187], and those for elliptic lower dimensionaltori only 15 years later [150, 347]. Theorem 17 treats all thenondegenerate typesof the normal behavior of invariant tori in a unified way, and Herman’s techniqueexploited in the proof of Theorem 19 enables one to explore non-hyperbolic lowerdimensional tori in individual systems as easily as hyperbolic ones.

By the way, in the hyperbolic case, the Center Manifold reduction allows oneto deduce Theorem 19 from the LagrangeanKAM theorem (with Russmann non-degeneracy conditions) in the finitely smooth setting, compare with§ 4.2. ForKolmogorov nondegeneracy, this approach was mentioned in [187] and discussedin detail in [220]. Finitely differentiable HamiltonianKAM Theory for ellipticlower dimensional tori was recently developed in [115].

All the unperturbed as well as perturbed invariant tori in Theorems 17 and 19are reducible, i.e., the variational equation along each of these tori canbe re-duced to a form with constant coefficients, see§ 5.2.3. The reducibility of theunperturbed tori is expressed by the fact that the matricesB in (46) and (53) and,consequently, the matricesΩ = JB in (47) and (54) arex-independent. How-ever, the HamiltonianKAM Theory for hyperbolic lower dimensional invarianttori can be generalized to non-reducible perturbed [187, 220] and even unper-turbed [125, 127, 264, 457] tori. In certain cases such a generalization can alsobe obtained for elliptic lower dimensional perturbed invariant tori [33]. The sep-aratrix stable and unstable manifolds (‘whiskers’) of the non-reducible perturbedhyperbolic tori are constructed in [187,457].

‘Exponential condensation’ as described in§ 8.1 also takes place at least insome lower dimensional settings in the analytic category [230,426].

For the reversible counterpart to the Hamiltonian lower dimensionalKAM The-ory, see [49, 66, 68, 269, 351, 378, 382, 384, 387, 389, 395, 443] as well as [48, 62,67, 381, 383, 400, 401]. These works also use quite differentnondegeneracy andnonresonance conditions on the unperturbed systems. For instance, Russmann-like nondegeneracy conditions for the unperturbed frequency map are assumed

79

in [67,68,389,395,400,401].

8.4 Excitation of elliptic normal modes

As was already pointed out, the hyperbolic case of the lower dimensional Hamilto-nianKAM settings treated in§ 8.3, where the2m×2m matricesΩ in (47) and (54)have no purely imaginary eigenvalues (N2 = 0), is generally believed to be muchsimpler than the non-hyperbolic case. However, consider more closely the lattercase, where the purely imaginary eigenvalues±iζ1, . . . ,±iζN2 of the matricesΩdo exist (N2 > 0). These eigenvalues are sometimes called theelliptic normalmodesof the unperturbed invariantn-tori Tn × y × 0 and yield the problemof the occurrence of Diophantine quasi-periodic invarianttori of dimensions fromn+1 to n+N2 near the surfaceΠ = z = 0 in the unperturbed systemX = XH

(or in the unperturbed family of systemsX = XHµ) as well as in perturbed sys-

temsX. If such tori of dimensionsn + 1, . . . , n + N2 exist, one sometimes saysthat the elliptic normal modes of the unperturbedn-tori Tn × y × 0 excite.

REMARKS.

1. Here we also suppose thatn > 2. The presence of quasi-periodic invarianttori of dimensionsκ from 2 to n + m (the number of degrees of freedom)near non-hyperbolic equilibria (n = 0) or periodic trajectories (n = 1) alsobelongs to the realm ofKAM Theory, of course, but this question is mucheasier and more ‘conventional’, and one usually does not speak of excita-tion of elliptic normal modes in this set-up. Instead, this topic is referredto as the ‘local’ LagrangeanKAM Theory (forκ = n + m) or the ‘local’lower dimensionalKAM Theory (forκ < n + m). By the way, ‘exponentialcondensation’ of invariantm-tori near elliptic equilibria of analytic Hamil-tonian systems withm degrees of freedom is established in [137,138].

2. In the Hamiltonian and reversible settings, the occurrence of purely imagi-nary normal eigenvalues is open (see the end of§ 8.3.1). In the general dis-sipative set-up, partial normal ellipticity is a codimension 1 phenomenon,meaning that it has a persistent occurrence in generic1-parameter familiesof systems, where it then takes place for isolated values of the parameter.The dissipative analogue of the present Hamiltonian discussion for non-hyperbolic invariantn-tori therefore is the branching off of(n + 1)-tori ina quasi-periodic Hopf bifurcation, as described in Section6. Here we alsokeep track of the internal and normal frequencies of then-torus, which pre-supposes the presence of sufficiently many parameters. In contrast, whenexamining excitation of elliptic normal modes in the Hamiltonian setting,we do not study any metamorphoses in the phase portrait of thesystem as

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the external parameter varies (even if the latter is present). In fact, thisexcitation can well occur in an individual system in the absence of any pa-rameters.

It turns out that under certain nondegeneracy and nonresonance conditions, in-variantκ-tori of all the dimensions in the rangen + 1 6 κ 6 n + N2 do existin the unperturbed system(s) as well as in perturbed systemswithin the frame-work of the lower dimensional HamiltonianKAM Theory. We shall not formulatethe corresponding theorems even vaguely and will confine ourselves with relevantreferences. Up to now, the phenomenon of excitation of elliptic normal modeshas been explored for analytic Hamiltonian systems only (although it undoubt-edly takes place forC∞- and finitely smooth systems as well). The first excitationresults were obtained in 1962–63 by V.I. Arnold [4,7] who considered the partic-ular caseκ − n = N2 = m. This case was recently revisited by M.R. Herman,see [97,160]. A.D. Bruno [91] examined the general case of arbitraryN2 6 m andκ 6 n+N2 and constructed analytic families of invariantκ-tori. General theoremsdescribing Whitney differentiable families of invariantκ-tori (for arbitraryN2 andκ) were proven in [68, 392] and independently in [231, 426] (see also [17, 232]).Analytic families of tori found by Bruno are subfamilies of these Whitney smoothfamilies. A. Jorba and J. Villanueva [231, 232, 426] also established the ‘expo-nential condensation’ of invariantκ-tori (which, of course, is impossible in thefinitely smooth andC∞-categories). Various versions of the excitation theoremhave been surveyed in detail in the review [393].

The phenomenon of the excitation of elliptic normal modes isalso known inreversible [68, 351, 385, 386, 389, 395] and volume preserving [398] set-ups. Theworks [68, 385, 389, 395] consider the excitation of elliptic normal modes in re-versible flows and the papers [351,386,389], in reversible diffeomorphisms.

8.5 Higher dimensional coisotropic invariant tori

Apart from theKAM Theory for lower dimensional isotropic invariant tori inHamiltonian systems we discussed in§ 8.3, there exists the ‘dual’ theory forhigher dimensional coisotropic invariant tori. The words ‘higher dimensional’mean that the dimensions of the tori are larger than the number of degrees of free-dom and the word ‘coisotropic’ means that the tangent spaceTpT to a torusT atany pointp ∈ T containsthe skew-orthogonal complement ofTpT with respectto the symplectic form (in the isotropic case, the spaceTpT is containedin itsskew-orthogonal complement).

The observation crucial for the higher dimensional HamiltonianKAM Theorywas made by M.R. Herman in 1988–89.

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LEMMA 20 (Automatic isotropicity) [210,211].Any quasi-periodic invariant torusof a Hamiltonian system is isotropic provided that the corresponding symplecticform is exact.

In fact, Herman [210,211] proved this lemma for a particularcase of invariantn-tori of symplectic diffeomorphisms of2n-dimensional symplectic manifolds,but the general case (verified in [68,160,399]) of invarianttori of arbitrary dimen-sions is not harder at all.

By the way, Lemma 20 implies that all the perturbed invarianttori in the ‘con-ventional’KAM Theory areautomaticallyisotropic (in particular, Lagrangean, iftheir dimension is equal to the number of degrees of freedom). This refers e.g. toTheorems 6, 16, 17, 19, and 21 (in the case of Theorem 21 below,the symplecticform is exact in a neighborhood of each torus).

Lemma 20 shows that the symplectic form in the higher dimensional KAM

Theory should be non-exact (compare also with Theorem 5 in [311]). Moreover,it turns out that the periods of the symplectic form (its integrals over the two-dimensional cycles within the tori in question) should satisfy certain Diophantine-like conditions: all the theorems on coisotropic tori proven by now include suchDiophantine hypotheses. The coisotropic HamiltonianKAM Theory was foundedby I.O. Parasyuk [337] in 1984, see also subsequent papers [242,243,338–340] byParasyuk and Kubichka (as well as the recent note [285]). Coisotropic invariantn-tori of Hamiltonian systems withN < n degrees of freedom were also studied byHerman [212,213] (see also [314,449,451,458]) and by F. Cong and Y. Li [126].

In Parasyuk’s theory, one starts with an unperturbed Hamiltonian system withN > 2 degrees of freedom whose phase space is smoothly foliated into coisotropicinvariantn-tori carrying conditionally periodic dynamics (N +1 6 n 6 2N − 1).Then, as in the Lagrangean casen = N we considered in§ 4.3, one can prove that,under certain conditions on the symplectic form and the unperturbed Hamiltonfunction, perturbed systems still admit many Diophantine quasi-periodic coisotropicinvariantn-tori. The proofs unavoidably involve Diophantine approximations ofdependent quantities, see§ 4.3.2. The measure of the complement to the unionof the perturbed tori, of course, vanishes as the perturbation size tends to zero.The symplectic form here is usually supposed to be fixed, as inthe ‘conventional’Lagrangean or lower dimensional isotropic HamiltonianKAM Theory. However,in their latest papers [243, 340], A.A. Kubichka and I.O. Parasyuk considered thecase where the symplectic form is perturbed as well (both theunperturbed andperturbed forms being assumed to meet certain Diophantine conditions).

Herman’s papers [212,213] (compare with [458]) are devotedto the particularcasen = 2N − 1. In this case, each unperturbed as well as perturbed energyhypersurface (to be more precise, each connected componentof each energy hy-persurface) is obviously ann-dimensional torus. It turns out that the frequency

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vectors of all the unperturbed tori in the set-up of [212, 213] are proportional tothe same Diophantine vectorω0 ∈ Rn and, moreover,all the perturbed energyhypersurfaces still carry Diophantine quasi-periodic dynamics with the frequencyvectorsalsoproportional to thesamevectorω0. This amazing picture (discussedalso in [68],§ 1.4.2) is a direct consequence of the fact that the Hamiltonian natureof a vector field imposes very severe restrictions on the motion on invariant tori ofsmall codimensions in the phase space.

The most important application of the coisotropic Hamiltonian KAM Theoryis counterexamples to the so-called Quasi-Ergodic Hypothesis [17, 68, 314, 449,451]: “on (almost) every compact and connected energy hypersurface of a genericHamiltonian system, there is an everywhere dense evolutioncurve”. The fal-lacy of this conjecture for the case of two degrees of freedomfollows from theLagrangeanKAM Theory (see§ 4.4.2): LagrangeanKAM 2-tori divide three-dimensional energy hypersurfaces and exclude everywhere dense evolution curves.For the case ofN > 3 degrees of freedom, exactly the same obstacle to quasi-ergodicity is provided by the coisotropicKAM Theory: coisotropic invariant(2N−2)-tori divide (2N − 1)-dimensional energy hypersurfaces and again exclude aneverywhere dense evolution curve: all the evolution curvesare forever trappedin between those tori. This reason is also due to Herman (see [314, 449, 451]).Besides, one obtains perpetual adiabatic stability of the two-dimensional ‘action’variable. However, whether the Quasi-Ergodic Hypothesis is valid for N > 3degrees of freedom and exact symplectic forms is still an open question.

8.6 Atropic invariant tori

Now suppose that a Hamiltonian systemX with N +m degrees of freedom (N >

2, m > 1) possesses a2N-dimensional normally hyperbolic invariant surfaceΠwith the following properties:

1. the restriction of the symplectic form toΠ is a symplectic form onΠ;

2. the induced Hamiltonian dynamics onΠ exemplifies the unperturbed sys-tem for some coisotropicKAM theorem (in the finitely smooth set-up).

In particular, the surfaceΠ is foliated intoX-invariantn-tori for somen in therangeN + 1 6 n 6 2N − 1 which are coisotropicwithin Π. In the ambientphase space, these tori are neither coisotropic nor isotropic: they are, as one says,atropic. Each Hamiltonian systemX sufficiently close toX will have a normallyhyperbolic invariant surfaceΠ that is close toΠ [117,162,215,424] and containsmanyX-invariant Diophantine quasi-periodicn-tori coisotropic withinΠ and at-ropic in the ambient phase space. The dimension of these torican be

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1. smaller than the number of degrees of freedom (ifm > 2 andN +1 6 n <N + minm, N);

2. equal to the number of degrees of freedom (if1 6 m 6 N − 1 andn =N + m);

3. greater than the number of degrees of freedom (ifN > 3, 1 6 m 6 N − 2,andN + m < n 6 2N − 1).

One concludes that the HamiltonianKAM Theory can be carried over to atropicinvariant tori. This was first observed by Q. Huang, F. Cong, and Y. Li [221,222]who in fact considered analytic systems and proved the existence of perturbedtori (with hyperbolic [221] as well as elliptic [222] normalbehavior) by entirelydifferent methods. The review [399] presents a detailed discussion of this new andpromising branch ofKAM Theory.

Note finally that the HamiltonianKAM Theory can be generalized to the casewhere the phase space is a Poisson manifold rather than a symplectic one [260,265, 270, 272]. Poisson manifolds are equipped with a Poisson bracket (of func-tions) which is allowed to be degenerate.

9 Whitney smooth bundles ofKAM tori

In classical mechanics many torus bundles are known to be nontrivial while the‘standard’KAM Theorem 6 of§ 4.3.1 only applies to trivial torus bundles of theform Tn×A. In this section we develop a globalKAM Theory, that can be appliedto nontrivial torus bundles, thereby obtaining a Cantorized bundle of which we canstill speak of nontriviality. Note that the problem of constructing such a theory wasmentioned by V.I. Arnold in a talk at the beginning of the 1990’s, who referred tothe paper [144].

9.1 Motivation

A motivation for globalizing the LagrangeanKAM theorem is the nontriviality ofcertain torus fibrations in Liouville integrable systems, for example in the spher-ical pendulum. Here an obstruction to the triviality of the fibration by Liouvilletori is given bymonodromy, see [129, 144, 145]. For a geometrical discussionof all obstructions for a toral fibration of an integrable Hamiltonian system to betrivial, see e.g. [28, 29, 144, 322, 327, 329]. A natural question is whether (non-trivial) monodromy also can be defined for non-integrable perturbations of e.g.the spherical pendulum. Answering this question is of interest in the study ofsemi-classical versions of such classical systems, see [131, 133, 305]. The results

84

discussed in this section imply that for an open set of Liouville integrable Hamil-tonian systems, under a sufficiently small perturbation, the geometry of the fibra-tion is largely preserved by a Whitney smooth diffeomorphism. Consequently,monodromy can be defined in the nearly integrable case as well[44, 50, 51]. Inparticular, our approach applies to the spherical pendulum. For a similar resultin the case of two degrees of freedom near a focus-focus singularity (or complexsaddle point), see [355]. We expect that a suitable reformulation of our resultswill be valid in the general Lie algebra setting of [67–69,223,312].

Our goal is to establish a global quasi-periodic stability result for fibrationsor bundles of Lagrangean tori, by gluing together local conjugations obtainedfrom the classical ‘local’KAM Theorem 6. This gluing uses a Partition of Unity[214, 318, 407] and the fact that invariant tori of the unperturbed integrable sys-tem have a natural affine structure [12, 51, 83, 129, 158], see§ 4.1.1. The globalconjugation is obtained as an appropriate convex linear combination of the localconjugations. This construction is reminiscent of the one used to build connec-tions or Riemannian metrics in differential geometry.

REMARK . In [51] it is also shown that the Whitney Extension Theorem [292,318,407,438] can be globalized to manifolds.

9.2 Formulation of the global KAM theorem

We now give a precise formulation of the globalKAM theorem [51] in the worldof C∞-systems. Consider a2n-dimensional, connected, smooth symplectic man-ifold (M, σ) with a surjective smooth mapπ : M → B, whereB is an n-dimensional smooth manifold. We assume that the mapπ defines a smooth lo-cally trivial fiber bundle, the fibers of which are Lagrangeann-tori. As before,Tn = Rn/(2πZ)n is the standardn-torus.

By the Liouville–Arnold Integrability Theorem [5, 12, 14, 17, 129, 297] it fol-lows that for everyb ∈ B there is a neighborhoodU b ⊆ B and a symplecticdiffeomorphism

ϕb : V b = π−1(U b) → Tn × Ab; m 7→(xb(m), yb(m)

),

with Ab ⊆ Rn an open set and with the symplectic form∑n

j=1 dxbj ∧ dyb

j onTn × Ab, such thatyb = (yb

1, yb2, . . . , y

bn) is constant on the fibers ofπ. We call

(xb, yb) angle-action variables and(V b, ϕb) an angle-action chart.Now consider a smooth Hamilton functionH : M → R, which is constant on

the fibers ofπ, that is,H is an integral ofπ. Then the corresponding Hamiltonianvector fieldXH defined byιXH

σ = dH is tangent to these fibers, for this notation

85

see [2,12,17,68,84,352] and§ 4.3.1 above. This leads to a vector field

(ϕb∗XH)(xb, yb) =

n∑

j=1

ωbj(y

b)∂

∂xbj

=n∑

j=1

∂(ϕb∗H)

∂ybj

∂

∂xbj

on Tn × Ab with the frequency vectorωb(yb) =(ωb

1(yb), . . . , ωb

n(yb)). We call

ωb : Ab → Rn the local frequency map. We say thatH is aglobally nondegenerateintegral ofπ (in the sense of Kolmogorov), if for a collection

(V b, ϕb)

b∈B

ofangle-action charts whose domainsV b coverM , each local frequency mapωb :Ab → Rn is a diffeomorphism onto its image.

REMARKS.

1. According to a remark of J.J. Duistermaat [144], there is anatural affinestructure on the space of actionsB. We note that, as in§ 4.1.1, the affinestructure is restricted somewhat further by the transitionmaps. Using thisaffine structure onB, for eachb ∈ B the second derivative ofhb =

(H|V b

)

(ϕb)−1 onTn ×Ab is well-defined. By global nondegeneracy we mean thatthe second derivativeD2hb has maximal rankn everywhere onV b for eachb ∈ B. Because of the affine structure onB, it follows that on overlappingangle-action charts the rank is independent of the choice ofchart.

2. Often the regularn-torus bundle is a part of a larger structure containingsingularities, see§ 3.2.2 for an example (the planar pendulum) and comparewith the Stefan–Sussmann theory [408,412], also see [28,29].

Suppose thatH is a globally nondegenerate integral onB. If B′ ⊆ B is arelatively compact subset ofB, that is, the closure ofB′ is compact, then there is afinite subcoverU bb∈F of U bb∈B′ such that for everyb ∈ F the local frequencymapωb is a diffeomorphism onto its image. Accordingly, letM ′ = π−1(B′) andconsider the corresponding bundleπ′ : M ′ → B′.

We shall take perturbations ofH in theC∞-topology onM [214, 318], for adescription see§ 4.1.2. We now formulate the global counterpart of Theorem 6.

THEOREM 21 (LagrangeanKAM for bundles) [51].Let (M, σ) be a smooth2n-dimensional symplectic manifold withπ : M → B a smooth locally trivial La-grangeann-torus bundle. LetB′ ⊆ B be an open and relatively compact subsetand letM ′ = π−1(B′). Suppose thatH : M → R is a smooth integral ofπ, whichis globally nondegenerate. Finally letF : M → R be a smooth function. IfF |M ′

is sufficiently small in theC∞-topology, then there is a subsetC ⊂ B′ and a mapΦ : M ′ → M ′ with the following properties.

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1. The subsetC ⊂ B′ is nowhere dense, and the measure ofB′ \ C tends to0as the size of the perturbationF tends to zero.

2. The subsetπ−1(C) ⊂ M ′ is a union of DiophantineXH-invariant La-grangeann-tori.

3. The mapΦ is a C∞-diffeomorphism onto its image and isC∞-near theidentity.

4. The restrictionΦ = Φ | π−1(C) conjugatesXH to XH+F , that is,

Φ∗XH = XH+F .

Note that the HamiltonianH + F need not be an integral ofπ.

REMARKS.

1. The mapΦ generally is not symplectic, compare with the local situation of§ 4.3.1.

2. Item 4 of Theorem 21 can be also expressed by saying thatXH is (globally)quasi-periodically stable onM ′. Note that, by the smoothness ofΦ, themeasure of the nowhere dense setΦ

(π−1(C)

), which is the union of the

perturbedn-tori, is large.

3. The restriction ofΦ to π−1(C) ⊂ M ′ preserves the affine structure of thequasi-periodic tori, see§§ 4.1, 4.2, and 4.3. In the complementM ′ \π−1(C)the diffeomorphismΦ hasnodynamical meaning. Still, the push forward ofthe integrable bundleπ′ : M ′ → B′ by Φ is a smoothn-torus bundle whichinterpolates the tori inΦ

(π−1(C)

).

4. Regarding the closed half line structure of the subbundleπ′ : π−1(C) → C,we can largely repeat the remarks made after Theorems 5 and 6.We like toadd that by the gluing of various angle-action charts, the half line structureis largely preserved [51]; one could speak here of aCantor manifold.

9.3 Applications

As said earlier in§ 9.1, many torus bundles, as these occur in Liouville integrableHamiltonian systems, are nontrivial. A notorious example is the spherical pen-dulum [129, 144], for more details see§ 9.3.1 below. Other examples are metin the Hamiltonian Hopf bifurcation, the champagne bottle,etc.; for an overviewsee [147].

87

I0

1

E

−1

Figure 7: Range of the energy-momentum map of the spherical pendulum.

We first explain the globalKAM Theorem 21 on the example of the sphericalpendulum, later on showing how monodromy can also be defined for Cantorizedtorus bundles as these occur in nearly integrable Hamiltonian systems. Also theconnection with quantum monodromy is briefly discussed.

9.3.1 Example: the spherical pendulum

Here we consider the spherical pendulum [12, 129, 144, 355].Dynamically, thespherical pendulum is the motion of a unit mass particle restricted to the unitsphere inR3 in a constant vertically downward gravitational field. The configura-tion space of the spherical pendulum is the2-sphereS2 =

q ∈ R3

∣∣ |q| = 1

andthe phase space, the cotangent bundle

M = T ∗ S2 =(q, p) ∈ R6

∣∣ |q| = 1 and〈q, p〉 = 0

of S2. Hereq = (q1, q2, q3) andp = (p1, p2, p3), while 〈·, ·〉, as usual, denotes thestandard inner product inR3.

The spherical pendulum is a Liouville integrable system. ByNoether’s Theo-rem [12, 17, 129] the rotational symmetry about the verticalaxis gives rise to theangular momentumI(q, p) = q1p2 − q2p1, which is a second integral of motion,in addition to the energyE = H(q, p) = 1

2|p|2 + q3. The energy-momentum map

of the spherical pendulum is

EM : T ∗ S2 → R2; (q, p) 7→ (I, E) =(q1p2 − q2p1,

12|p|2 + q3

).

Its fibers corresponding to regular values give rise to a fibration of the phase spaceby Lagrangean2-tori. The imageB of EM is the closed part of the plane lying inbetween the two curves meeting at a corner, see Figure 7. The set of singular val-ues ofEM consists of the two boundary curves and the points(I, E) = (0,±1).These points correspond to the equilibria(q, p) = (0, 0,±1, 0, 0, 0), whereas the

88

boundary curves correspond to the horizontal periodic motions of the pendulumdiscovered by Huygens [224]. Therefore the setB of regularEM-values consistsof the interior ofB minus the point(I, E) = (0, 1), corresponding to the unstableequilibrium point(0, 0, 1, 0, 0, 0). This point is the center of the nontrivial mon-odromy. The corresponding fiberEM−1(0, 1) is a once pinched2-torus. Note thatEM : M → B is a singular foliation in the sense of Stefan–Sussmann [408,412].

OnB, one of the two components of the frequency map is single valued whilethe other is multi-valued [129, 144]. E. Horozov [219] established global nonde-generacy ofH onB. Thus the globalKAM Theorem 21 can be applied to any rela-tively compact open subsetB′ ⊆ B. Consequently, the integrable dynamics on the2-torus bundleEM′ : M ′ → B′ of the spherical pendulum is quasi-periodicallystable.

This means that any sufficiently small Hamiltonian perturbation of the spher-ical pendulum has a nowhere dense union of Diophantine invariant tori of largemeasure, which admits a smooth interpolation by a push forward of the integrablebundleEM′ : M ′ → B′. In § 9.3.2 we shall argue that this allows for a definitionof (nontrivial) monodromy for the perturbed torus bundle.

What precisely happens when the spherical pendulum system is slightly per-turbed within the world of Hamiltonian systems? First, whenthe perturbationpreserves the axial symmetry, the perturbed system remainsLiouville integrableby Noether’s Theorem [12,17,129] and hence the monodromy ispreserved [300,328].

The question now is what happens to the monodromy when by the perturbationthe axial symmetry is broken, thereby yielding a nearly integrable system.

9.3.2 Monodromy in the nearly integrable case

Here we indicate how to define the concept of monodromy for a nearly integrablenowhere dense torus bundle [51]. Our construction, however, is independent ofany integrable approximation.

A regular union of tori. Let M be a manifold endowed with a (smooth) metric%. Then for any two subsetsA, B ⊆ M we define%(A, B) = infx∈A,y∈B %(x, y).Note that in general this does not define a metric on the set of all subsets:%(A, B) =0 does not implyA = B, and%(A, C) can well exceed%(A, B) + %(B, C). LetM ′ ⊆ M be compact andTλλ∈Λ a collection of pairwise disjointn-tori in M .We require the followingregularity properties. There exist positive constantsεandδ such that

1. For allλ ∈ Λ, each continuous maphλ : Tλ → Tλ with %(x, h(x)

)< 2ε

for all x ∈ Tλ, is homotopic with the identity mapIdTλ;

89

2. For eachλ, λ′ ∈ Λ, such that%(Tλ, Tλ′) < δ, there exists a homeomorphismhλ′,λ : Tλ → Tλ′ , such that%

(x, hλ′,λ(x)

)< 1

2ε for all x ∈ Tλ;

3. For eachx ∈ M ′ there existsλ ∈ Λ such that%(x, Tλ

)< 1

2δ.

Note that the homeomorphismhλ′,λ, as required to exist by item 2, by item 1 isunique modulo homotopy. We also observe the following. In the present situation,we started with a globally nondegenerate Liouville integrable Hamiltonian sys-tem, such that an open and dense part of its phase spaceM is foliated by invariantLagrangeann-tori. Let M ′ be a compact union of such tori, then, under suffi-ciently small, non-integrable perturbation, the remaining LagrangeanKAM tori inM ′ as discussed in this section, form a regular collection in the above sense.

Construction of a Zn-bundle. We now construct aZn-bundle, first only overthe unionM ′′ =

⋃λ Tλ, which is assumed to be regular in the above sense. For

each pointx ∈ Tλ the fiber is defined byFx = H1(Tλ, Z) ≈ Zn, the first homol-ogy group ofTλ ≈ Tn overZ. Using the regularity property 2 and the fact thathλ′,λ is unique modulo homotopy, it follows that this bundle is locally trivial. LetE ′′ denote the total space of this bundle andπ′′ : E ′′ → M ′′ the bundle projection.

Now this bundle is extended overM ′ ∪ M ′′ as follows. For eachx ∈ M ′ wedefine

Λ(x) =λ ∈ Λ

∣∣ %(x, Tλ

)< 1

2δ.

Then we consider the set of pairs(λ, α), whereλ ∈ Λ(x) andα ∈ H1(Tλ, Z). Onthis set we have the following equivalence relation:

(λ, α) ∼ (λ′, α′) ⇐⇒ (hλ′,λ)∗α = α′,

whereh∗ denotes the action ofh on the homology. The set of equivalence classesis defined as the fiberFx atx. The fiberFx is isomorphic toFx′ in a natural (andunique) way for anyx′ ∈ Tλ with λ ∈ Λ(x). This extended bundle again is locallytrivial.

We conclude by observing that the monodromy of the initial torus bundle ex-actly is the obstruction to global triviality of theZn-bundle just constructed. More-over, by the globalKAM Theorem 21, in the integrable case one obtains the sameZn-bundle as in the nearly integrable case.

9.4 Discussion

We described global quasi-periodic stability for bundles of Lagrangean invarianttori, under the assumption of global Kolmogorov nondegeneracy on the integrableapproximation. We emphasize that our approach works for arbitrarily many de-grees of freedom and that it is independent of the integrablegeometry one starts

90

with. The global Whitney smooth conjugationΦ between Diophantine tori of theintegrable and the nearly integrable systems can be suitably extended to a smoothmapΦ, which serves to smoothly interpolate the nearly integrable, only Whitneysmooth Diophantine torus bundleΦ

(π−1(C)

), defined over the nowhere dense set

C. We observe that if the extended diffeomorphisms are sufficiently close to theidentity, they are also isotopic to the identity, see [51]. Therefore the interpolationbecomes bundle isomorphic with its integrable counterpartπ′ : M ′ → B′. Thismeans that the global geometry is as in the integrable case.

In this sense, the lack of unicity of Whitney extensions plays no role and wecan generalize concepts like monodromy directly to the nearly integrable case.For a topological discussion of the correspondingn-torus bundles, see [144, 322,327–329].

The present global quasi-periodic stability result directly carries over to thegeneral setting of [67–69, 223, 312]. Within the world of Hamiltonian systems,this leads to applications at the level of lower dimensionalisotropic tori, see§ 8.3. However, the approach also applies to dissipative, volume preserving, orreversible systems, compare with [48,49,66].

REMARK . The asymptotic considerations of B.W. Rink [355] near focus-focussingularities (i.e., complex saddles) in Liouville integrable Hamiltonian systemswith two degrees of freedom, allow for a similar applicationto the spherical pen-dulum whereB′ is a small annular region around the point(I, E) = (0, 1). Theconclusion again is that the nearly integrable systems havenontrivial monodromy.

It is tempting to combine the globalKAM Theory with that of quasi-periodicbifurcations [48, 54]. In the ensuing Cantorization, apartfrom closed half lines,also higher dimensional closed half spaces do occur. It is tobe noted that inthe quasi-periodic reversible Hopf bifurcation, also nontrivial monodromy occurs[48].

One of the main motivations for the interest in nontriviality of symplectic torusbundles in Hamiltonian systems, is the connection with certain spectral propertiesin related semi-classical systems. In particular this refers to the so-called spectraldefect. This connection is proven by Vu Ngo.c San [427]. Many applications havebeen reported in [130,131,146–149,428], for overviews see[44,131,409].

By Theorem 21 and its consequences, classical monodromy also exists in thenearly integrable case. As far as we know, quantum monodromyonly is well-defined for integrable systems, often obtained by just truncating a nearly inte-grable system. Similarly in a number of the Quantum Theory applications, theclassical limit is not integrable, but only nearly so. It is an open question whetherquantum monodromy also can be defined in nearly integrable cases, but we expectthat the present approach may be useful in this.

91

10 Conclusion

The lasting influence of Kolmogorov, Arnold, and Moser on thepresent state ofthe art in Mathematics, Physics and other sciences is enormous and this reviewonly sketches part of this legacy. Nevertheless we believe that it is an importantpart, which is still fully in development. The role of the Ergodic Hypothesis inStatistical Mechanics has turned out to be much more subtle than was expected,see e.g. [36, 171, 172]. RegardingKAM Theory, for further reading we mentionthe introductory texts [109, 123, 274, 296, 349]. Also reading of [17, 68, 155, 248,257,259,275,399] is recommended. The discussion around quantum monodromyand its relationship toKAM Theory, in particular to the Cantorization of Whitneysmooth bundles, has not yet come to its conclusion [51,131,147,409]. The role ofquasi-periodic bifurcation theory [58,68,69,199,432,433] and the correspondingCantorization and fraying undoubtedly will become increasingly more relevant,especially since now higher dimensional modelling is practiced more often.

For general reference we also refer to the companion Handbooks [163, 201,202].

The authors thank Richard Cushman, Konstantinos Efstathiou, Aernout van En-ter, Francesco Fasso, Heinz Hanßmann, Jun Hoo, George Huitema, Sergeı Kuksin,Olga Lukina, Jan van Maanen, Anatoliı Neıshtadt, Hinke Osinga, Joaquim Puig,Bob Rink, Khairul Saleh, Carles Simo, Floris Takens, Dmitriı Treshchev, RenatoVitolo, and Florian Wagener for helpful discussions. The first author is grateful toJurgen Moser for his guidance inKAM Theory. The second author is very muchobliged to Vladimir Igorevich Arnold who taught himKAM Theory. Moreover,the first author acknowledges hospitality of the Universitat de Barcelona and theUniversite de Bourgogne and partial support of the Dutch FOM program Mathe-matical Physics (MF-G-b) and of European Community fundingfor the Researchand Training Network MASIE (HPRN-CT-2000-00113). The second author isgrateful to the University of Maryland for its hospitality and acknowledges partialsupport of the Council for Grants of President of the Russia Federation, GrantsNo. NSh-1972.2003.1 and NSh-4719.2006.1.

92

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