Singularity Theory for Non-twist Tori · PDF fileIntroduction Introduction Alex Haro (UB)...

Singularity Theory for Non-twist Tori

A. González A. Haro R. de la Llave

Universitat de Barcelona & University of Texas at Austin

Hamiltonian Intensive Seminar IIFacultat de Matemàtiques i Estadística, UPC

December 3, 2010

Alex Haro (UB) Singularity Theory for Non-twist Tori UPC 2010 1 / 48

Index

1 Introduction

2 Motivating examples

3 Methodology

4 Some applications

5 Interactions with Symplectic Geometry

6 Conclusions


Introduction

Introduction


Introduction

KAM TheoryExistence and persistence of quasi-periodic motions in Hamiltonian systems

Under twist conditions (' diffeomorphic frequency map):

Persistence of quasi-periodic motions (Kolmogorov, Arnold,Moser...)

A posteriori results of existence of invariant tori with fixedfrequency, based on reducibility (de la Llave, González,Jorba, Villanueva)

Existence of Cantor families of tori under weaker conditions:Unfoldings of quasi-periodic tori (Broer, Huitema, Moser,Pöschel, Sevryuk, Takens, ...)

Persistence of invariant tori under Rüssmann condition(Sevryuk, Rüssmann, Xu, You, Zhang, ...)


Introduction

Non-twist tori with fixed frequencyPrevious results

Numerical results in non-twist area preserving maps:del Castillo-Negrete, Greene, Morrison (1996, 1997):periodic orbits and transition to chaos (Apte, Wurm, Petrisor,Shinohara, Aizawa, ...)

Haro (2002): computation of non-twist tori and normal forms.

Rigorous results:Simó (1998): existence of meandering curves in areapreserving non-twist maps (but the curves are twist!).

Delshams, de la Llave (2002): existence and persistence ofnon-twist curves in 2-parameter families of apm.

Dullin, Ivanov, Meiss (2006): normal forms for the fold andcusp singularities for 4D maps.


Introduction

Non-twist tori with fixed frequencyThis talk

Singularity Theory for non-twist tori(arbitrary dimension and any finite-determined singularity)

Bifurcations of invariant tori, with fixed frequency.At the bifurcation points, the invariant tori are non-twist.

Classification of the possible degeneracies of invariant tori.

Unfolding Hamiltonian systems around non-twist tori.

Persistence of non-twist invariant tori.

Efficient numerical methods to study bifurcations.


Motivating examples

Motivating examples


Motivating examples

Motivation 1: Integrable systemsFrequency map of an integrable symplectomorphism

Consider the integrable system f0 : Tn × Rn → Tn × Rn,

f0(x , y) =

(x +∇W (y)

y

),

where W : Rn → R.

For p ∈ Rn, the torus Kp(θ) =

(θp

)is invariant for f0 with

frequency ω(p) = ∇W (p):

f0(Kp(θ)) = Kp(θ + ω(p)) .

Kp is twist if the torsion T (p) = Dω(p) = Hess W (p) is non-degenerate.

Observation: non-twist tori (with fixed frequency) can bedestroyed, even with integrable perturbations.


Motivating examples

Motivation 1: Integrable systemsInvariant tori with frequency ω

Goal: Look for invariant tori with frequency ω.

Define the modified family

fλ(x , y) = f0(x , y) +

(λ0

).

A torus K (θ) is fλ-invariant with frequency ω if and only if:

fλ(K (θ)) = K (θ + ω) .

For p ∈ Rn, the couple (λ(p),Kp(θ)) with counterterm

λ(p) = ω −∇W (p)

satisfies the invariance equation.Alex Haro (UB) Singularity Theory for Non-twist Tori UPC 2010 9 / 48

Motivating examples

Motivation 1: Integrable systemsInvariant tori as critical points of the potential A

Define the potential A(p) = W (p)− p>ω.Hence:

λ(p) = −∇pA(p).T (p) = Hess A(p).

Then:

Kp∗ is an f0-invariant torus with frequency ω if and only if p∗is a critical point of the potential A(p).

Kp∗ is a twist f0-invariant torus with frequency ω if and onlyp∗ is a non-degenerate critical point of the potential A(p).


Motivating examples

Motivation 1: Integrable systemsThe simplest degenerate example

Assume A(p) =p3

3ω(y) = ω + y2.

p = 0 is a degenerate critical point of A(p).

The torus K0(θ) is non-twist.

Unfolding: A(p;µ) = µp +p3

3ω(y ;µ) = ω + µ+ y2.


Motivating examples

Motivation 2: A collision of invariant toriA family of quadratic standard maps

Consider the quadratic standard family of symplectomorphismsfκ : T× R→ T× R, defined by:

x = 0.375 + x + y2 ,

y = y − κ

2πsin(2πx) .

Problem: Look for invariant tori with frequency ω = 3−√

52 , with

respect to parameter κ.


Motivating examples

Motivation 2: A collision of invariant toriNumerical observations

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

y

x

K= 0.000000

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

y

x

K= 1.000000

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

y

x

K= 1.361408

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

x

K= 1.500000


Motivating examples

Questions(and partial answers)

In non-integrable cases, can we define an action whosecritical points correspond to invariant tori of a fixed ω?We construct a family of translated tori, so that the invariant toricorrespond to translated tori whose translation is zero. The translationis Moser’s counterterm.

Is there a general theory of bifurcations of KAM tori?The counterterm is a gradient function, i.e. with a potential. Hence, weuse Singularity Theory for analyzing the critical points of the potential.

Can we obtain results of persistence of non-twist tori (offixed frequency) under perturbations?Singularity Theory suggests that “finitely determined” non-twist tori arepersistent in families of a high enough number of parameters.


Methodology

Methodology


Methodology

MethodologySetting

Let fκ : Tn × Rn → Tn × Rn be a smooth family of exactsymplectomorphisms, with κ ∈ P ⊂ Rm:

Dfκ(x , y)>J Dfκ(x , y) = J, where J =

(On −InIn On

);

There exists a primitive function Sκ : Tn × Rn → R, such thatdSκ = f ∗κ (y dx)− y dx .

Let ω ∈ Rn be fixed and Diophantine (ω ∈ Dn(γ, τ)):∣∣`>ω −m∣∣ ≥ γ |`|− τ1 , ∀` ∈ Zn \ 0, m ∈ Z.

Goal: Finding fκ-invariant tori, with frequency ω.


Methodology

MethodologyModified family and parameter-torus deformations

1) Consider the modified family

f(κ,λ)(x , y) = fκ(x , y) +

(λ0

),

where −λ ∈ Rn is the counterterm.

2) Use KAM techniques to find a family of parameter-torus

K : D× P×Tn −→ Λ×Tn×Rn

(p, κ, θ) −→ (λ(p;κ),K(p;κ)(θ)),

such that:f(κ,λ(p;κ)) K(p;κ)(θ) = K(p;κ)(θ + ω) ,⟨

K(p;κ)(θ)− Kp(θ)⟩

= 0 .

Important: The nondegeneracy condition is very mild.Alex Haro (UB) Singularity Theory for Non-twist Tori UPC 2010 17 / 48

Methodology

MethodologyPotential and countertem

3) Define the potential

A(p;κ) = −p>λ(p;κ)−⟨

Sκ K(p;κ)

(θ)⟩,

where Sκ is the primitive function of fκ.

The following holds:

λ(p;κ) = −∇pA(p;κ)

Hence, fixed κ = κ∗:

K(p∗;κ∗)

is fκ∗-invariant with frequency ω if and only if p∗ is acritical point of the potential A(p;κ∗).


Methodology

MethodologyPotential and torsion

4) The torsion of K(p;κ)

is the n × n symmetric matrix

T (p;κ) =⟨

N(p;κ)

(θ + ω)> J Dzf(κ,λ(p;κ))(K(p;κ)(θ)) N

(p;κ)(θ)⟩,

where N(p;κ)

(θ) = J DθK(p;κ)(θ)(

DθK(p;κ)(θ)>DθK(p;κ)

(θ))−1

.

There exist n × n matrices W1(p;κ) and W2(p;κ) such that:

T (p;κ) W1(p;κ) = W2(p;κ) HessA(p;κ) .

Hence, if W1(p∗;κ∗) and W2(p∗;κ∗) are invertible:

The co-rank of p∗ as a critical point of A(p;κ∗) equals theco-rank of the torsion T (p∗;κ∗).


Some applications

Some applications


Some applications

Example 1: A persistence resultHypotheses

Theorem (Persistence theorem of non-twist tori)Given:

ω ∈ Rn, a Diophantine frequency vector.f0, a real-analytic integrable simplectic map with frequencymap ω(y) = ω +∇A(y), s.t. y∗ = 0 is a finitely determinedsingularity of A(y).fµ, a µ-family of real-analytic integrable simplectic map withfrequency map ωµ(y) = ω +∇Aµ(y), s.t. Aµ(y) is a stableunfolding of the singularity at zero.fµ,ε, a perturbed family of exact symplectomorphims:

fµ,ε(x , y) =

(x + ω +∇Aµ(y)

y

)+ εg(x , y , ε).


Some applications

Example 1: A persistence resultTheses

Theorem (Theses)Then:

(KAM) For (µ, ε) sufficiently small, there exists a potentialAµ,ε(p) defined around 0 ∈ Rn and an n-dimensional torusKp,µ,ε : Tn → Tn × Rn, depending smoothly on (p, µ, ε), suchthat:

fµ,ε Kp,µ,ε(θ) = Kp,µ,ε(θ + ω) +

(∇pAµ,ε(p)

0

).

(ST) For ε sufficiently small, there exist µ(ε) and p(ε) suchthat Aµ(ε),ε has a singularity at p(ε) that is of the same type(in the sense of Singularity Theory) of the singularity of A0

at zero.Alex Haro (UB) Singularity Theory for Non-twist Tori UPC 2010 22 / 48

Some applications

Example 1: A persistence resultSketch of proof

f (x , y) =

(x + ω +∇A(y)

y

)Unfold the singularity

yfµ(x , y) =

(x + ω +∇Aµ(y)

y

)Perturb

yfµ,ε(x , y) =

(x + ω +∇Aµ(y)

y

)+ εg(x , y , ε).


Some applications

Example 1: A persistence resultSketch of proof

1 For each p, Kp,µ,0 = Kp is an approximate translated torusfor fµ,ε with translation −λµ(p) = ∇Aµ(p):

fµ,ε(Kp(θ)) = Kp(θ + ω)−(λµ(p)

0

)+ εg(Kp(θ), ε).

2 Apply the methodology to construct a family of couples

(λµ,ε(p),Kp,µ,ε), invariant under fµ,ε +

(λ0

), parameterized by

the moment:⟨K y

p,µ,ε⟩

= p.3 Use symplectic geometry to define the potential Aµ,ε(p).4 Apply Singularity Theory to obtain perturbed degenerate

tori.Alex Haro (UB) Singularity Theory for Non-twist Tori UPC 2010 24 / 48

Some applications

Example 2: A collision of toriApplying the methodology

Modified family:

f(κ,λ)(x , y) =

(0.375 + x +

(y − κ

2π sin(2πx))2

+ λy − κ

2π sin(2πx)

).

The primitive function of fκ,λ is

Sκ(x , y) =κ

4 π2 cos(2πx) +23

(y − κ

2πsin(2πx)

)3.

The co-rank of the torsion of a torus is either 0 (twist) or 1(non-twist).


Some applications

A fold singularityA collision of invariant tori

-0.2

-0.15

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-0.05

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

y

x

K= 0.000000

T = 0.1669253T = −0.1669253

-0.005

0

0.005

0.01

-0.1 -0.05 0 0.05 0.1p

-0.0004

-0.0002

0

0.0002

0.0004

-0.1 -0.05 0 0.05 0.1

A

p


Some applications


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-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

y

x

K= 1.000000

T = 0.1123464T = −0.1123464

-0.005

0

0.005

0.01

-0.1 -0.05 0 0.05 0.1p

-0.0004

-0.0002

0

0.0002

0.0004

-0.1 -0.05 0 0.05 0.1

A

p


Some applications


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-0.15

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-0.05

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

y

x

K= 1.361408

T = 0.6558 · 10−8

T = −0.6558 · 10−8

-0.005

0

0.005

0.01

-0.1 -0.05 0 0.05 0.1p

-0.0004

-0.0002

0

0.0002

0.0004

-0.1 -0.05 0 0.05 0.1

A

p


Some applications


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0

0.05

0.1

0.15

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

x

K= 1.500000

-0.005

0

0.005

0.01

-0.1 -0.05 0 0.05 0.1p

-0.0004

-0.0002

0

0.0002

0.0004

-0.1 -0.05 0 0.05 0.1

A

p


Some applications

Example 3: The birth of meandering toriAnother family of quadratic standard maps

Consider a quadratic standard family of symplectomorphismsfκ : T× R→ T× R, defined by:

x = x + (y + 0.1)(y − 0.2) ,

y = y − κ

2πsin(2πx) .

Problem: Look for invariant tori with frequency ω =√

5−132 , with

respect to parameter κ.


Some applications

A fold singularityThe birth of meandering tori

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0.1

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

x

K= 0.420000

-0.02

-0.01

0.00

0.01

0 0.02 0.04 0.06 0.08 0.1p

-0.0022

-0.0020

-0.0018

-0.0016

0 0.02 0.04 0.06 0.08 0.1

A

p


Some applications


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0

0.1

0.2

0.3

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0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

x

K= 0.430396

T = −1.6261 · 10−7

T = 1.6261 · 10−7

-0.02

-0.01

0.00

0.01

0 0.02 0.04 0.06 0.08 0.1p

-0.0022

-0.0020

-0.0018

-0.0016

0 0.02 0.04 0.06 0.08 0.1

A

p


Some applications


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

x

K= 0.440000

T = −0.0211805T = 0.0211805

-0.02

-0.01

0.00

0.01

0 0.02 0.04 0.06 0.08 0.1p

-0.0022

-0.0020

-0.0018

-0.0016

0 0.02 0.04 0.06 0.08 0.1

A

p


Some applications


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y

x

K= 0.450000

T = −0.0309003T = 0.0309003

-0.02

-0.01

0.00

0.01

0 0.02 0.04 0.06 0.08 0.1p

-0.0022

-0.0020

-0.0018

-0.0016

0 0.02 0.04 0.06 0.08 0.1

A

p


Interactions with Symplectic Geometry

Interactions withSymplectic Geometry



Compatible triple in an annulusDefinition and matrix representation

A ⊂ Tn × Rn is an annulus, endowed with a compatible tiple

(ω = dα,J,g) .

The matrix representation of the compatible triple is

(Ω(z) = Da(z)> − Da(z), J(z),G(z)) ,

where a : A → Rn, that satisfy the relations

Ω> = −Ω, J2 = −I2n, G> = G,

Ω = J>ΩJ = GJ = −J>G, G = J>GJ = −ΩJ = J>Ω .



Geometric properties ofinvariant Lagrangian toriAutomatic reducibility

Setting: A is an annulus with a compatible triple (ω= dα,J,g),and A0 ⊂ A is another annulus.

Lemma (Automatic reducibility)Let f : A0 → A be a (homotopic to the identity)symplectomorphism: f ∗ω = ω.Let K : Tn → A0 be a (homotopic to the zero-section) torus.Assume that K is f -invariant and that the dynamics is theergodic rotation Rω(θ) = θ + ω: f K = K Rω.Then, the following hold:

a) f is exact symplectic: f ∗α−α = dS.b) K is Lagrangian: K ∗ω = 0.

HermanAlex Haro (UB) Singularity Theory for Non-twist Tori UPC 2010 37 / 48


Lemma (cont.)

c) Define LK ,NK : Tn → R2n×n and MK : Tn → R2n×2n by

MK (θ) = (DθK (θ)︸︷︷︸LK (θ)

J(K (θ)) DθK (θ) GK (θ)−1︸︷︷︸NK (θ)

) ,

where GK (θ) = LK (θ)>G(K (θ))LK (θ). Then, the vectorbundle morphism induced by MK :

MK : Tn × R2n −→ TKA0

(θ, ξ) −→ (K (θ),MK (θ)ξ)

is an isomorphism such that M∗Kω = ω0.



Lemma (cont.)d) The linearized dynamics Dz f K reduces to block-triangular:

MK (θ + ω)−1Dz f (K (θ))MK (θ) =

(In T

(f ,K )(θ)

On In

),

where T(f ,K )

is the symmetric n×n matrix defined by

T(f ,K )

(θ) = NK (θ + ω)> Ω(K (θ + ω)) Dz f (K (θ)) NK (θ) .



Geometric properties ofinvariant fibered Lagrangian deformationsHamiltonian deformations

DefinitionLet Λ ⊂ Rs be open. A Hamiltonian deformation with base Λ is asmooth function g : Λ×A0 → A inducing a family of exactsymplectomorphisms:

g : Λ −→ Sympe(A0,A)λ −→ gλ.

A primitive function of g is a smooth function S : Λ×A0 → Asuch that g∗λα−α = Sλ.



Geometric properties ofinvariant fibered Lagrangian deformationsMoment map of a Hamiltonian deformation

The generator of g is G : Λ×A0 → R2n×s defined by

G(λ, z) = Dλg(λ,g−1λ (z)).

The moment map of g is G : Λ×A0 → Rs defined by

G(λ, z)> = a(z)>G(λ, z)− DλS(λ,g−1λ (z)).

The potential of g is V : Λ×A0 → R defined by

V (λ, z) = −G(λ, z)>λ− S(λ,g−1λ (z)).

Important: G(λ, z) = Ω(z)−1DzG(λ, z)>.Alex Haro (UB) Singularity Theory for Non-twist Tori UPC 2010 41 / 48


Geometric properties ofinvariant fibered Lagrangian deformationsExistence of a potential

DefinitionLet D,Λ ⊂ Rs be open.

i) A fibered Lagrangian deformation (FLD) is a smooth map

K = (λ,K ) : D×Tn −→ Λ×A0

(p, θ) −→ (λ(p),Kp(θ)),

such that, for each p ∈ D, Kp ∈ Lag(Tn,A0).ii) The FLD K = (λ,K ) is g-invariant with frequency ω if for any

p ∈ D, gλ(p) Kp − Kp Rω = 0.iii) The FLD K = (λ,K ) is parameterized by the momentum

parameter p if G(p) := 〈G(λ(p),Kp(θ))〉 = p.Alex Haro (UB) Singularity Theory for Non-twist Tori UPC 2010 42 / 48


Theorem (The potential of a FLD)Let g : Λ×A0 → A be a Hamiltonian deformation.Let K = (λ,K ) : D×Tn → Λ×A0 be a g-invariant FLD, withfrequency ω ∈ Dn(γ, τ), with momentum parameter p.Define V : D → R as V (p) = 〈V (λ(p),Kp(θ))〉.Then, the following holds:

λ(p) = −∇pV (p).

T (p) Dpc(p) = W (p) Hessp V (p),

where:T (p) =

⟨T(gλ(p),Kp)(θ)

⟩is the torsion of Kp w.r.t. gλ(p);

c(p) =⟨a(Kp(θ))>DθKp(θ)

⟩> is the averaged action of Kp;

W (p) = 〈DzG(λ(p),Kp(θ)) (Np(θ)− Lp(θ)RωTp(θ)) 〉>, withRωTp(θ)−RωTp(θ+ω) = Tp(θ)−〈Tp(θ)〉 and 〈RωTp(θ)〉 = 0.


Conclusions

Conclusions


Conclusions

Conclusions

The infinite dimensional problem of finding invariant tori forHamiltonian systems is reduced to the finite dimensionalproblem of finding critical points of the potential.

Non-twist tori correspond to degenerate critical points of thepotential.

The Singularity Theory for invariant tori is provided by theSingularity Theory of the critical points of the potential.

There is an analogous theory for Hamiltonian vector fields.


Conclusions

Conclusions

The method yields very efficient numerical algorithms tocompute invariant tori and its bifurcations.

The method is suitable to validate numerical computations.

We are able to study any finite-determined singularity of thefrequency map.

The method can be also applied for obtaining small twisttheorems.

Our method is designed for the cases on which numericalevidence of bifurcations is known but the system is notclose to integrable.


A road map

Symplectic deformations and moment maps Reducibility of invariant tori

?

Approx. reducibility of approx. invariant tori

?

Potential for invariant fibered Lagrangian deformations ?

Parametric KAM Theorem

? ?

Existence of invariant fibered Lagrangian deformations

? ?

Singularity Theory for invariant tori

??Persistence of invariant tori forclose-to-integrable systems

Non-twist tori forclose-to-integrable systems

Numerical methods forinvariant tori bifurcations

?

@@@@R

?


Thanks!


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