Complex Dynamics of Some Hamiltonian Systems...

Research ArticleComplex Dynamics of Some Hamiltonian SystemsNonintegrability of Equations of Motion

Jingjia Qu 12

1School of Mathematics Jilin University Changchun 130012 China2Department of Basic Science Aviation University Air Force Changchun 130022 China

Correspondence should be addressed to Jingjia Qu qujingjia-123163com

Received 6 November 2018 Revised 23 January 2019 Accepted 11 February 2019 Published 3 March 2019

Academic Editor Sandro Wimberger

Copyright copy 2019 Jingjia Qu This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Themain purpose of this paper is to study the complexity of someHamiltonian systems from the view of nonintegrability includingthe planar Hamiltonian with Nelson potential double-well potential and the perturbed elliptic oscillators Hamiltonian Somenumerical analyses show that the dynamic behavior of these systems is very complex and in fact chaotic in a large range of theirparameter I prove that these Hamiltonian systems are nonintegrable in the sense of Liouville My proof is based on the analysis ofnormal variational equations along some particular solutions and the investigation of their differential Galois group

1 Introduction

The topic of integrability and nonintegrability for systems ofnonlinear differential equations is one of the central problemsin the qualitative theory of ODEs A system can be regardedas integrable if it admits such a number of first integrals orother tensor invariants that it is solvable by quadratures Theexistence of 119896 functionally independent first integrals canhelp us reduce the study of the considered 119899-dimensionalsystem to (119899 minus 119896)-dimensional one Restricted on the levelsets of first integrals we can understand the complexity andtopological structure of the considered dynamical systemFor example if a Hamiltonian system with 119899 degrees offreedom is integrable in the sense of Liouville ie it has 119899functionally independent first integrals in involution thenthe invariant sets associated with first integrals are genericallydiffeomorphic to tori cylinders or planes inside the phasespace If the system is not integrable we can expect it mayexhibit a variety of chaotic phenomena or complex dynamicalbehavior

In general it is difficult to determine whether a systemis integrable or not see [1 2] for instance For all we knowthe first results on the nonintegrability should go back to

Poincare [3] who found the influence of resonances ofthe eigenvalues of the monodromy matrix associated to aperiodic orbit for small generic perturbations of integrablesystems Then in the study of the rigid body with one fixedpoint [4] Kovalevskaya showed that a given system whosesingularities are only poles turns out to be integrable insome sense The idea of Kovalevskaya was the first trying todetect the integrability of dynamical systems by studying thesingular structure of their general solutions

Ziglin [5 6] investigated the existence of meromorphicfirst integrals for analytic Hamiltonian systems Based onproperties of the monodromy group of a normal variationalequation (NVE) along a complex integral curve he presentednecessary conditions for a 2119899-dimensional complex analyticHamiltonian system to have 119899 functionally independentrational first integrals It should be pointed out that Ziglinrsquostheorem does not deal with the integrability in the Liouvillesense of the Hamiltonian system with arbitrary numberof degrees of freedom in his original formulation sincethe considered first integrals may not be in involutionHowever since the Poisson bracket of first integrals and theHamiltonian always vanishes Ziglinrsquos results can be appliedto study nonintegrability of the Hamiltonian system with

HindawiAdvances in Mathematical PhysicsVolume 2019 Article ID 9326947 10 pageshttpsdoiorg10115520199326947

2 Advances in Mathematical Physics

two degrees of freedom such as Toda lattice Hamiltonian[7] Kepler problems [8] Lie-Poisson system [9] and Euler-Poisson system [6]

An important progress was made by Morales RamisSimo [10 11] and Baider Churchill Rod and Singer [12] inthe end of 20th centuryThey replaced themonodromy groupof NVEwith the differential Galois group of NVE and showedthat the Liouville integrability of the nonlinear Hamiltoniansystem implies the integrability of the linear system NVE inthe sense of expressibility of solutions in closed form

Consider an analytic Hamiltonian system 119883119867 on a com-plex analytic symplectic manifold119872 and the equation of themotion for119867 reads119889119889119905119911 = 119883119867 119911 = (1199111 119911119899) isin C2119899 119905 isin C (1)

The variational equation (VE) along a particular solution 119911 =120601(119905) of (1) has the form of119889119889119905120585 = 119860 (119905) 120585 119860 (119905) = 120597119883119867120597119911 (120601 (119905)) 120585 isin C2119899 (2)

By using the linear first integral 119889119867(120601(119905)) of VE(2) (2) can bereduced into the NVE119889119889119905120578 = 119861 (119905) 120578 120578 isin C2(119899minus1) (3)

in suitable coordinates [10]

Theorem 1 (see [10]) Let 119867 be a Hamiltonian in C2119899 andlet Γ be a particular solution Assume that there exist 119899rational first integrals of 119883119867 (1) which are in involution andfunctionally independent in a neighborhood of Γ Then theidentity component of the differential Galois group of NVE(3)is Abelian

Remark 2 As was pointed in Ref [10] if the variational equa-tion is Fuchsian ie means that all singularities includinginfinity are regular one can extend the class of first integralsand get the meromorphic version of Theorem 1

Compared with Ziglinrsquos theorem the Morales-Ramistheory is more effective to study the nonintegrability ofthe Hamiltonian systems On the one hand the differentialGalois group is bigger than the monodromy group andthere are linear differential equationswith trivialmonodromygroup but with nontrivial differential Galois group On theother hand the differential Galois group is an algebraicgroup in particular a Lie group and can be calculated byinfinitesimal methods Since then Morales-Ramis theory hasbeen applied successfully for studying the nonintegrabilityof large numbers of physical models such as the planarthree-body problem [13ndash17] Hillrsquos problem [18] generalizedYang-Mills Hamiltonian [19] Wilberforce spring-pendulumproblem [20] and double pendula problem [21] It shouldbe pointed out that the differential Galois group can also beused to investigate the nonintegrability of general dynamicalsystems which may be non-Hamiltonian [22ndash24]

The aim of this paper is to investigate some Hamil-tonian systems including Nelson Hamiltonian a double-well potential Hamiltonian and perturbed elliptic oscillators

Hamiltonian By using Morales-Ramis theory and Kovacicrsquosalgorithm we will contribute to the understanding of thecomplexity or the topological structure of these systems fromthe point of view of nonintegrability

In order to apply theMorales-Ramis theory it is necessaryto know how to check if the identity component of thedifferential Galois group of a given linear system is AbelianGenerally speaking this is a hard work However effectivemethods to investigate this question for lower dimensionalsystems with rational coefficients do exist In particularfor second order equations over C(119909) the algorithm ofKovacic [25] allows to decidewhether the identity componentof the differential Galois group is solvable Based on theconsiderations above we can solve the problems in this paperaccording to the following steps

Firstly finding a nonconstant particular solution Γ ofthe considered Hamiltonian system Secondly calculatingthe variational equation along Γ furthermore we can getthe normal variational equation Then we transform thenormal variational equation to an equation with rationalcoefficients by using a transformation which does not changethe identity component of the differential Galois groupFinally we investigate the differential Galois group of theequation with rational coefficients with the help of Kovacicrsquosalgorithm

2 Main Results

21 Nonintegrability of Nelson Hamiltonian Consider theHamiltonian119867 = 12 (1199012119909 + 1199012119910) + (119910 minus 121199092)2 + 12058311990922 (4)

where 119909 119910 isin C 120583 is an arbitrary real parameter Its associatedHamiltonian system is = 119901119909119910 = 119901119910119909 = 119909 (2119910 minus 1199092 minus 120583) 119910 = 1199092 minus 2119910

(5)

This differential system is known as the Hamiltonian systemwith Nelson potential Interest in it is coming from nuclearphysics where the deep valley could represent a collectivedegree of freedom which remains coupled to other types ofexcitation [26]

The Nelson Hamiltonian resembles the Henon-Heilespotential and it also has a rich well-known periodic orbitstructure [27] Furthermore the motion which it generatescan be shown to be bounded and has been widely studied interms of periodic orbits [28] bifurcations [29] and quantummechanically [30 31]

The main goal of this section is to study the noninte-grability of system (5) by using Morales-Ramis theory andKovacicrsquos algorithm Our result is formulated in the followingtheorem

Advances in Mathematical Physics 3

Theorem 3 The system (5) is nonintegrable in the sense ofLiouville

Proof of Theorem 3 It is easy to find that the manifold

N = (119909 119910 119901119909 119901119910) isin C4 | 119909 = 119901119909 = 0 (6)

is invariant with respect to the flow of (5) and system (5) hasa solution Γ 119909 (119905) = 0119910 (119905) = cosradic2119905119901119909 (119905) = 0119901119910 (119905) = minusradic2 sinradic2119905

(7)

which lies in N Let (1205851 1205852 1205781 1205782) be variations in(119909 119910 119901119909 119901119910) Then the variational equation of (4) along Γreads

( 1205851120585212057811205782)=(0 0 1 00 0 0 12 cosradic2119905 minus 120583 0 0 00 minus2 0 0)(

1205851120585212057811205782) (8)

Since the particular solution lies on N the equation forvariables (1205851 1205781) forms a closed subsystem( 12058511205781) = ( 0 12 cosradic2119905 minus 120583 0)(12058511205781) (9)

which is the so-called normal variational equation and isequivalent to 1205851 = (2 cosradic2119905 minus 120583) 1205851 (10)

Making a change of variable 119911 = cosradic2119905 we can transform(10) into 120585101584010158401 = 119886 (119911) 12058510158401 + 119887 (119911) 1205851 (11)

The prime denotes the derivative with respect to 119911 where119886 (119911) = 1199111 minus 1199112 119887 (119911) = 2119911 minus 1205832 (1 minus 1199112) (12)

Putting 1205851 = 119883 exp 12 int119886 (120591) 119889120591 (13)

then (11) can be changed to the reduced form11988310158401015840 = 119903 (119911)119883 (14)

where119903 (119911) = minus 316 (119911 minus 1)2 minus 316 (119911 + 1)2 minus 4120583 + 916 (119911 + 1)+ 4120583 minus 716 (119911 minus 1) (15)

In what follows we will show that the differential Galoisgroup of (14) is 119878119871(2C) Clearly the set of singularities ofequation (14) isP = minus1 1infin the order at 119911 = minus1 and 119911 = 1is two and the order at 119911 = infin is one necessary conditionsfor Cases 1 and 3 are not satisfied and only Case 2 should beanalysed see Theorem A2 in Appendix Indeed by simplecomputation we get 119864infin = 1 119864minus1 = 2 1 3 1198641 = 2 1 3 (16)

However for any 119890infin isin 119864infin 119890minus1 isin 119864minus1 1198901 isin 1198641 119889 =(12)(119890infin minus 119890minus1 minus 1198901) le (12)(1 minus 1 minus 1) lt 0 so Case 2 inTheorem A2 in Appendix cannot hold too ByTheorem A2in Appendix the differential Galois group 119866 of (14) is in Case4 so the corresponding identity component of (14) is also119878119871(2C)

Assume system (5) is integrable in the sense of Liouvillethen by Theorem 1 we conclude that the identity componentof the differential Galois group of (10) is Abelian Note thatthe transformation 119911 = cosradic2119905 does not change the identitycomponent of this group (page 31Theorem 25 [10])Thus theidentity component of (11) is Abelian Since an Abelian groupis always a solvable group and the solvability of (11) coincideswith that of (14) we show that the identity component of(14) is solvable However 119878119871(2C) is not solvable We cancomplete the proof of this theorem

We remark that (10) is a particular case of MathieuEquation and its solvability has been studied by P Acosta-Humanez et al [32] In their works they also investigated alarge class of Hamiltonians with rational potentials which isclose to Nelson potential

By using Symplectic algorithm and Matlab one canobserve a rich variety of behavior of the Nelson system inPoincare cross section as the energy increases For examplewe consider the Poincare cross section with 119910 = 001Figure 1 shows that the section changes from regular patternto irregular pattern as the energy level increases from 001 to015 this indeed implies the phenomena of chaos Theorem 3shows that (5) is nonintegrable We all know that lack of firstintegrals can imply a complex behavior of phase curves of thesystem and the numerical result just illustrates this point

22 Nonintegrability of a Double-Well Potential HamiltonianNext we consider the planar Hamiltonian with a double-wellpotential119867 = 12 (1199012119909 + 1199012119910) minus 1205721199102 exp (minus120573 (1199092 + 1199102)) (17)


0

H=005

minus06 minus04 minus02 0 02 04 06 08minus08x

minus04

minus03

minus02

minus01

01

02

03

04

px

H=001

minus05 minus03 minus02 minus01 0 01 02 03 04 05minus04x

minus02

minus015

minus01

minus005

0

005

01

015

px

H=015

minus08

minus06

minus04

minus02

0

02

04

06

px


0

H=01

minus05

minus04

minus03

minus02

minus01

01

02

03

04

05

px


Figure 1 The Poincare cross sections of the Nelson Hamiltonian system on the surface 119910 = 001 when 120583 = 005where 120572 120573 are real parameters Its associated Hamiltonianequation is 119909 = 119901119909 = 119901119910119909 = minus21205721205731199091199102 exp (minus120573 (1199092 + 1199102)) 119910 = 2120572119910 (1 minus 1205731199102) exp (minus120573 (1199092 + 1199102))

(18)

The motivation for the choice of the potential 119881 =minus1205721199102 exp(minus120573(1199092 + 1199102)) comes from the interest of thispotential in chaotic scattering [33 34] This system can beconsidered as a model to describe the electron scattering onthe 119867+2 ion or the prototype for the trapping of incomingstellar objects by a collection ofmassive objects such as a solarsystem [35]

Numerical tests in [35] show that the motion of system(18) has sensitive dependence on initial conditions andchaotic scattering occurs when 120572 = 120573 = 1 Howeversuch tests do not exclude the possibility that the system

(18) admits complex or chaotic behavior for other valuesof (120572 120573) In this work from a view of the integrability westudy the nonintegrability of system (18) and contribute tounderstanding the complexity of system (18)

In the case of 120572120573 = 0 obviously the Hamiltonian system(18) is integrable Suppose 120572120573 = 0 we proceed to prove thetheorem below

Theorem 4 For 120572120573 = 0 the Hamiltonian system (18) isnonintegrable in the sense of Liouville

Proof of Theorem 4 It is easy to see that the manifold

M = (119909 119910 119901119909 119901119910) isin C4 | 119909 = 119901119909 = 0 (19)

is invariant with respect to the flow generated by system (18)Indeed system (18) restricted toM is given by119910 = 119901119910119910 = minus2120572119910 (1205731199102 minus 1) exp (minus1205731199102) (20)


which is completely integrable with the first integral 119867|M =(12)1199012119910 minus 1205721199102 exp(minus1205731199102) Fixing the level set 119867|M = 0 weobtain a particular solution (119909 119910 119901119909 119901119910) = (0 119910(119905) 0 (119905))with 1199102 = 21205721199102 exp(minus1205731199102) and denote by Γ the phase curvecorresponding to this solution The variational equation of(18) along the solution (0 119910(119905) 0 (119905)) is given by

( 1205851120585212057811205782)=( 0 0 1 00 0 0 1minus21205721205731199102 exp (minus1205731199102) 0 0 00 119872 0 0)(

1205851120585212057811205782)(21)

where 119872 = (412057212057321199104 minus 101205721205731199102 + 2120572) exp(minus1205731199102) The cor-responding normal variational equation reads( 12058511205781) = ( 0 1minus21205721205731199102 exp (minus1205731199102) 0)(12058511205781) (22)

or equivalently 1205851 = (minus21205721205731199102 exp (minus1205731199102)) 1205851 (23)

The right-hand side of this equation is nontrivial provided120572 = 0 Making a change of variable 119911 = 1199102(119905) then (23)becomes 120585101584010158401 = 119886 (119911) 12058510158401 + 119887 (119911) 1205851 (24)

and 119886 (119911) = 120573119911 minus 22119911 119887 (119911) = minus 1205734119911 (25)

Furthermore let 1205851 = 119883 exp 12 int119886 (120591) 119889120591 (26)

Then (24) becomes 11988310158401015840 = 119903 (119911)119883 (27)

where 119903 (119911) = 120573216 minus 1205732119911 minus 141199112 (28)

Then we show that the differential Galois group of (27)is 119878119871(2C) It follows fromTheorem A1 in Appendix that weneed only to check that Cases 1 2 and 3 cannot hold Due

to 120573 = 0 we see that the function 119903(119911) admits two singularpoints 1199111 = 0 with order two and 1199112 = infin with order zeroWe can easily know that Case 3 inTheorem A2 cannot holdMoreover the difference of exponents at 119911 = 0 is zero Thenin a neighbourhood of 119911 = 0 two independent local solutionshave the form1205961 (119911) = 11991112119891 (119911) 1205962 (119911) = 1205961 (119911) ln (119911) + 11991112119892 (119911) (29)

where 119891(119911) and 119892(119911) are analytic at 119911 = 0 and 119891(0) = 0It follows that the differential Galois group can be only fulltriangular of 119878119871(2 119862) (for details see [36 37]) Hence onlyCase 1 or 4 of the Kovacicrsquos algorithm is possible

For Case 1 a simple computation leads to[radic119903]0 = 0120572+0 = 120572minus0 = 12 (30)

and

[radic119903]infin = 100381610038161003816100381612057310038161003816100381610038164 ] = 0119886 = 100381610038161003816100381612057310038161003816100381610038164 119887 = minus1205732 120572plusmninfin isin 1 minus1

(31)

It is not difficult to check that for any 119904(0) 119904(infin) isin + minus119889 = 120572119904(infin)infin minus120572119904(infin)0 is not a nonnegative integerTherefore Case1 cannot hold

Based on above discussion we can make the conclusionthat Case 4 in Theorem A1 in Appendix holds namely thedifferential Galois group of (27) is 119878119871(2C) Using the sameargument as in the proof ofTheorem3we can easily completethe proof of this theorem

23 Nonintegrability of Perturbed Elliptic Oscillators Hamilto-nian We study the following Hamiltonian which appearedin [38] 119867 = 12 (1199092 + 1199102 + 1199112 + 1199012119909 + 1199012119910 + 1199012119911)+ 120576 (11990921199102 + 11990921199112 + 11991021199112 minus 119909211991021199112) (32)

where 120576 is an arbitrary real parameter The Hamiltonian weconsider above consists of three coupled harmonic oscillators


known as perturbed elliptic oscillators Its associated Hamil-tonian system is = 119901119909119910 = 119901119910 = 119901119911119909 = minus119909 minus 120576 (21199091199102 + 21199091199112 minus 211990911991021199112) 119910 = minus119910 minus 120576 (21199092119910 + 21199101199112 minus 211990921199101199112) 119911 = minus119911 minus 120576 (21199092119911 + 21199102119911 minus 211990921199102119911)

(33)

The reason for the choice of this Hamiltonian is thatperturbed elliptic oscillators appear very often in galacticdynamics and atomic physics [39ndash41] Perturbed ellipticoscillators display exact periodic orbits interesting stickyorbits together with large chaotic regions [42] We want toshow that the system there may cause complex behavior fromthe view of nonintegrability

If 120576 = 0 it is easy to get the conclusion that theHamiltonian system (33) is integrable If 120576 = 0 we obtain thefollowing theorem

Theorem 5 Assume 120576 = 0 then the system (33) is noninte-grable in the sense of Liouville

Proof of Theorem 5 Obviously system (33) has a periodicsolution Γ 119909 (119905) = 0119910 (119905) = 0119911 (119905) = sin 119905119901119909 (119905) = 0119901119910 (119905) = 0119901119911 (119905) = cos 119905

(34)

It is easy to check that the variational equation of (32) alongΓ is(((((((

120585112058521205853120578112057821205783)))))))

=((((((

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1minus1 minus 2120576 sin2119905 0 0 0 0 00 minus1 minus 2120576 sin2119905 0 0 0 00 0 1 0 0 0))))))((((((

120585112058521205853120578112057821205783))))))(35)

and corresponding normal variational equation is

( 1205851120585212057811205782)=( 0 0 1 00 0 0 1minus1 minus 2120576 sin2119905 0 0 00 minus1 minus 2120576 sin2119905 0 0)(

1205851120585212057811205782)(36)

Making a change of the location of the variable we get thefollowing equation

( 1205851120578112058521205782)=( 0 1 0 0minus1 minus 2120576 sin2119905 0 0 00 0 0 10 0 minus1 minus 2120576 sin2119905 0)(

1205851120578112058521205782)(37)

Clearly the normal variational equation (37) along Γ isa system of two uncoupled second order linear differentialequations 1205851 = minus (1 + 2120576 sin2119905) 1205851 (38)1205852 = minus (1 + 2120576 sin2119905) 1205852 (39)

We denote by 1198661 1198662 and 119866 the differential Galois groupof the equations (38) (39) and (37) respectively over the fieldof rational functions on Γ As a representation of an elementof 119866 is of the form (119860 00 119861) (40)

where 0 is the 2 times 2 null matrix 119860 isin 1198661 and 119861 isin 1198662the identity component of 119866 is not Abelian if the identitycomponent of 1198661 or 1198662 is not Abelian Then we turn toconsider the normal variational equation (38) and analyze thedifferential Galois group 1198661

In order to do that we make the usual change of variable119911 = sin2119905 then (38) becomes120585101584010158401 = 119886 (119911) 12058510158401 + 119887 (119911) 1205851 (41)

where 119886 (119911) = 2119911 minus 12119911 minus 21199112 119887 (119911) = minus 1 + 21205761199114119911 minus 41199112 (42)


Introducing the change of variable1205851 = 119883 exp 12 int119886 (120591) 119889120591 (43)

we have that the equation (41) becomes11988310158401015840 = 119903 (119911)119883 (44)

where119903 (119911) = minus 38119911 minus 3161199112 + 3 + 41205768 (119911 minus 1) minus 316 (119911 minus 1)2 (45)

This theorem will follow fromTheorem 1 if we can show thatthe differential Galois group of (44) is 119878119871(2C)

Note that the set of singularities of equation (44) is P =0 1infin The order at 119911 = 0 and 119911 = 1 is 2 It follows from120576 = 0 that the order at 119911 = infin is 1 We now proceed as in theproof ofTheorem 3 ByTheoremA2 in Appendix Cases 1 and3 do not occur

For Case 2 by simple computation we get119864infin = 1 1198640 = 2 1 3 1198641 = 2 1 3 (46)

However 119889 = (12)(119890infin minus 119890minus1 minus 1198901) le (12)(1 minus 1 minus 1) lt 0it is not a nonnegative integer so Case 2 cannot happen ByTheorem A2 in Appendix we know that Cases 1 2 and 3 inTheorem A1 in Appendix cannot hold we have thus provedthe theorem

Appendix

A Kovacicrsquos Algorithm

Consider the differential equation12058510158401015840 = 119903120585 119903 isin C (119909) (A1)

where C(119909) is the field of rational functions defined on thecomplex plane C

We recall that any general second order linear differentialequation can be transformed to the form (A1) by change ofvariable Indeed for equation11991010158401015840 = 1198861199101015840 + 119887119910 119886 119887 isin C (119909) (A2)

if we make the change of variable 119910 = 119890(12) int 119886120585 then (A2)becomes 12058510158401015840 = (11988624 minus 11988610158402 + 119887) 120585 (A3)

To avoid triviality we assume that 119903 is not a constantThen one has the following

TheoremA1 (see [25]) There are four cases that can occur

Case 1 119866 is conjugate to a triangular group Then (A1) has asolution of the form 119890int120596 with 120596 isin C(119909)Case 2 119866 is not of Case 1 but it is conjugate to a subgroup of119863 = (119888 00 119888minus1) | 119888 isin C 119888 = 0cup ( 0 119888minus119888minus1 0) | 119888 isin C 119888 = 0 (A4)

Then (A1) has a solution of the form 119890int120596 with 120596 algebraic overC(119909) of degree 2Case 3 119866 is not of Cases 1 and 2 but it is a finite group Thenall solutions of (A1) are algebraic over C(119909)Case 4 119866 = 119878119871(2C) Then (A1) is not integrable in Liouvillesense

Furthermore corresponding to the four cases listed inTheorem A1 there are some easy necessary conditions forCases 1 2 and 3 which consequently form a sufficientcondition for Case 4 Since 119903 isin C(119909) it can be representedas 119903 = 119904119905 with 119904 119905 isin C[119909] relatively prime Then the poles of119903 are the zeros of 119905 and we mean the order of the pole by themultiplicity of the zero of 119905 By the order of 119903 atinfin we shallrefer the order ofinfin as a zero of 119903 ie deg 119905 minus deg 119904 Then wehave the following

Theorem A2 (see [25]) The following conditions are neces-sary for the respective cases of Theorem A1 to hold

Case 1 Every pole of 119903 must have even order or else have order1 The order of 119903 atinfinmust be even or else be greater than 2

Case 2 119903 must have at least one pole that either has odd ordergreater than 2 or else has order 2

Case 3 The order of a pole of 119903 cannot exceed 2 and the orderof 119903 atinfinmust be at least 2 If the partial fraction expansion of119903 is 119903 = sum

119894

120572119894(119909 minus 119888119894)2 +sum119895 120573119895119909 minus 119889119895 (A5)

then radic1 + 4120572119894 isin Q for each 119894 and sum119895 120573119895 = 0 Furthermore ifwe let 120574 = sum

119894120572119894 +sum119895120573119895119889119895 (A6)

then radic1 + 4120574 isin Q

In what follows we describe the Kovacicrsquos algorithm [25]LetP be the set of singularities of linear differential equation(A1) containing the poles of 119903(119911) andinfin

A1 Case 1 of Kovacicrsquos Algorithm


Step 1 For each 119888 isin P we define [radic119903]119888 120572plusmn119888 as follows(a) If 119888 is a pole of order 1 then [radic119903]119888 = 0 120572+119888 = 120572minus119888 = 1(b) If 119888 is a pole of order 2 then [radic119903]119888 = 0 120572plusmn119888 = 12 plusmn(12)radic1 + 4119887 where 119887 is coefficients of 1(119909minus119888)2 in the partial

fraction expansion for 119903(c) If 119888 is a pole of order 2] ge 4 then [radic119903]119888 = 119886(119909 minus119888)] + sdot sdot sdot + 119889(119909 minus 119888)2 of negative order part of the Laurent

series expansion of radic119903 at 119888 120572plusmn119888 = (12)(plusmn(119887119886) + ]) where 119887is the coefficient of 1(119909 minus 119888)]+1 in 119903 minus the coefficient of1(119909 minus 119888)]+1 in [radic119903]2119888

(d) If the order of 119903 atinfin is gt 2 then [radic119903]infin = 0 120572+infin = 0120572minusinfin = 1(e) If the order of 119903 atinfin is 2 then [radic119903]infin = 0 120572plusmninfin = 12plusmn(12)radic1 + 4119887 where 119887 is coefficients of 11199092 in the Laurent

series expansion of 119903 atinfin(f) Ifinfin is a pole of order minus2] le 0 then [radic119903]infin = 119886119909] +sdot sdot sdot + 119889 of positive order part of the Laurent series expansion

ofradic119903 atinfin 120572plusmninfin = (12)(plusmn(119887119886)+ ]) where 119887 is the coefficientof 119909]minus1 in 119903minus the coefficient of 119909]minus1 in [radic119903]2infinStep 2 Let 119889 = sum119888isinP 119905(119888)120572119904(119888)119888 where 119904(119888) isin + minus for any119888 isin P 119905(infin) = + and 119905(119888) = minus for any 119888 isin P infin If 119889 is anonnegative integer then let 119908 = sum119888isinP(119904(119888)[radic119903]119888 +120572119904(119888)119888 (119909minus119888)) otherwise the family is discarded If no families remainunder consideration Case 1 of Theorem A1 cannot happen

Step 3 For each family retained from Step 2 we search fora monic polynomial 119875 of degree 119889 such that the equation11987510158401015840 + 21199081198751015840 + (1199081015840 + 1199082 minus 119903)119875 = 0 holds If such a polynomialexists then 120585 = 119875119890int119908 is a solution of 12058510158401015840 = 119903120585 If no suchpolynomial is found for any family retained from Step 2 Case1 of Theorem A1 cannot happen


Step 1 For each 119888 isin P we define 119864119888 as follows(a) If 119888 is a pole of order 1 then 119864119888 = 4(b) If 119888 is a pole of order 2 then 119864119888 = 2 + 119896radic1 + 4119887 119896 =0 plusmn2 cap Z where 119887 is coefficients of 1(119909 minus 119888)2 in the partial

fraction expansion for 119903(c) If 119888 is a pole of order ] gt 2 then 119864119888 = ](d) If the order of 119903 atinfin is gt 2 then 119864infin = 0 2 4(e) If the order of 119903 atinfin is 2 then119864infin = 2+119896radic1 + 4119887 119896 =0 plusmn2capZ where 119887 is coefficients of 11199092 in the Laurent series

expansion of 119903 atinfin(f) Ifinfin is a pole of order ] lt 2 then 119864infin = ]

Step 2 Let 119889 = (12)sum119888isinP 119905(119888)119890119888 where 119890119888 isin 119864119888 for any119888 isin P 119905(infin) = + and 119905(119888) = minus for any 119888 isin P infin If119889 is a nonnegative integer then let 120579 = (12)sum119888isinP 119890119888(119909 minus 119888)otherwise the family is discarded If no families remain underconsideration Case 2 of Theorem A1 cannot happen

Step 3 For each family retained from Step 2 we search for amonic polynomial 119875 of degree 119889 such that the equation 119875101584010158401015840 +312057911987510158401015840 + (31205792 + 31205791015840 minus 4119903)1198751015840 + (12057910158401015840 + 31205791205791015840 + 1205793 minus 4119903120579 minus 21199031015840)119875 = 0holds If such a polynomial exists let 120601 = 120579 + 1198751015840119875 and let 119908be a solution of the equation 1199082 + 120601119908 + ((12)1206011015840 + (12)1206012 minus

119903) = 0 then 120585 = 119890int119908 is a solution of 12058510158401015840 = 119903120585 If no suchpolynomial is found for any family retained from Step 2 Case2 of Theorem A1 cannot happen


Step 1 For 119899 isin 4 6 12 fixed we define 119864119888 with 119888 isin P asfollows

(a) If 119888 is a pole of order 1 then 119864119888 = 12(b) If 119888 is a pole of order 2 then 119864119888 = 6 +(12119896119899)radic1 + 4119887 119896 = 0 plusmn1 plusmn2 plusmn(1198992) cap Z where 119887 is

coefficients of 1(119909 minus 119888)2 in the partial fraction expansion for119903(c)119864infin = 6+(12119896119899)radic1 + 4119887 119896 = 0 plusmn1 plusmn2 plusmn(1198992)cap

Z where 119887 is coefficients of 11199092 in the Laurent seriesexpansion of 119903 atinfin

Step 2 Let 119889 = (11989912)sum119888isinP 119905(119888)119890119888 where 119890119888 isin 119864119888 for any119888 isin P 119905(infin) = + and 119905(119888) = minus for any 119888 isin P infin If 119889is a nonnegative integer then let 120579 = (11989912)sum119888isinP 119890119888(119909 minus 119888)119878 = prod119888isinPinfin(119909 minus 119888) otherwise the family is discarded If nofamilies remain under consideration Case 3 of Theorem A1cannot happen

Step 3 For each family retained from Step 2 we search fora monic polynomial 119875 of degree 119889 such that the recursiveequations 119875119899 = 119875119875119894minus1 = minus119878119875119894 + ((119899 minus 119894) 1198781015840 minus 119878120579) 119875119894minus (119899 minus 119894) (119894 + 1) 1198782119903119875119894+1(119894 = 119899 119899 minus 1 0) (A7)

with 119875minus1 equiv 0 hold If such a polynomial exists let 119908 be asolution of the equation sum119899119894=0(119878119894119875119894(119899 minus 119894))119908119894 = 0 then 120585 =119890int119908 is a solution of 12058510158401015840 = 119903120585 If no such polynomial is foundfor any family retained from Step 2 Case 3 of Theorem A1cannot happen

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The author declares having no conflicts of interest

Acknowledgments

This paper is supported by ESF grant (No JJKH20170776KJ)of Jilin China

References

[1] J J Morales-Ruiz ldquoMeromorphic nonintegrability of Hamilto-nian systemsrdquo Reports on Mathematical Physics vol 48 no 1-2pp 183ndash194 2001


[2] K Gozdziewski A J Maciejewski and Z Niedzielska ldquoSearch-ing for integrable systems An application of Kozlov theoremrdquoReports on Mathematical Physics vol 46 no 1-2 pp 175ndash1822000

[3] V V Kozlov ldquoIntegrability and non-integrability in Hamilto-nian mechanicsrdquo Russian Mathematical Surveys vol 38 no 1pp 1ndash76 1983

[4] S Kowalevski ldquoSur le probleme de la rotation drsquoun corps solideautour drsquoun point fixerdquoActaMathematica vol 12 no 1 pp 177ndash232 1889

[5] S L Ziglin ldquoBranching of solutions and the nonexistence of firstintegrals in Hamiltonian mechanicsIrdquo Functional Analysis andIts Applications vol 16 pp 181ndash189 1982

[6] S L Ziglin ldquoBranching of solutions and the nonexistence of firstintegrals in Hamiltonian mechanics IIrdquo Functional Analysisand its Applications vol 17 no 1 pp 6ndash17 1983

[7] H Yoshida ldquoNonintegrability of the truncated Toda latticeHamiltonian at any orderrdquo Communications in MathematicalPhysics vol 116 no 4 pp 529ndash538 1988

[8] H Yoshida ldquoNonintegrability of a class of perturbed Keplerproblems in two dimensionsrdquo Physics Letters A vol 120 no 8pp 388ndash390 1987

[9] A J Maciejewski and K Gozdziewski ldquoNumerical evidenceof nonintegrability of certain lie-poisson systemrdquo Reports onMathematical Physics vol 44 no 1-2 pp 133ndash142 1999

[10] J Morales-Ruiz Differential Galois Theory and Nonintegrabilityof Hamiltonian Systems vol 179 of Progress in MathematicsBirkhauser 1999

[11] JMorales-Ruiz and C Simo ldquoPicard-vessiot theory and Ziglinrsquostheoremrdquo Journal of Differential Equations vol 107 no 1 pp140ndash162 1994

[12] A Baider R C Churchill D L Rod and M Singer ldquoOn theinfinitesimal geometry of integrable systemsrdquo inMechanics Day(Waterloo ON 1992) vol 7 of Fields Institute Communicationsvol 7 pp 5ndash56 American Mathematical Society ProvidenceRI USA 1996

[13] A V Tsygvintsev ldquoThe meromorphic non-integrability of thethree-body problemrdquo Journal fur die Reine und AngewandteMathematik vol 537 pp 127ndash149 2001

[14] D Boucher and J-A Weil ldquoApplication of J-J Morales andJ-P Ramisrsquo theorem to test the non-complete integrability ofthe planar three-body problemrdquo in From Combinatorics toDynamical Systems C Mitschi and F Fauvet Eds vol 3 ofIRMA Lectures in Mathematics andTheoretical Physics pp 163ndash177 Walter De Gruyter Berlin Germany 2003

[15] A V Tsygvintsev ldquoNon-existence of new meromorphic firstintegrals in the planar three-body problemrdquoCelestial Mechanicsand Dynamical Astronomy vol 86 no 3 pp 237ndash247 2003

[16] P B Acosta-Humanez M Alvarez-Ramırez and J DelgadoldquoNon-integrability of some few body problems in two degreesof freedomrdquoQualitativeTheory of Dynamical Systems vol 8 no2 pp 209ndash239 2009

[17] A J Maciejewski and M Przybylska ldquoNon-integrability ofthe three-body problemrdquo Celestial Mechanics and DynamicalAstronomy vol 110 no 1 pp 17ndash30 2011

[18] J J Morales-Ruiz C Simo and S Simon ldquoAlgebraic proofof the non-integrability of Hillrsquos problemrdquo Ergodic Theory andDynamical Systems vol 25 no 4 pp 1237ndash1256 2005

[19] S Shi and W Li ldquoNon-integrability of generalized Yang-Mills Hamiltonian systemrdquoDiscrete and Continuous DynamicalSystems vol 33 no 4 pp 1645ndash1655 2013

[20] P B Acosta-Humanez M Alvarez-Ramırez D Blazquez-Sanz and J Delgado ldquoNon-integrability criterium for normalvariational equations around an integrable subsystem andan example the wilberforce spring-pendulumrdquo Discrete andContinuous Dynamical Systems vol 33 no 3 pp 965ndash986 2013

[21] T Stachowiak and W Szuminski ldquoNon-integrability ofrestricted double pendulardquo Physics Letters A vol 379 no47-48 pp 3017ndash3024 2015

[22] W Li and S Shi ldquoGaloisian obstruction to the integrability ofgeneral dynamical systemsrdquo Journal of Differential Equationsvol 252 no 10 pp 5518ndash5534 2012

[23] M Ayoul and N T Zung ldquoGaloisian obstructions to non-Hamiltonian integrabilityrdquo Comptes Rendus MathematiqueAcademie des Sciences Paris vol 348 no 23-24 pp 1323ndash13262010

[24] P B Acosta-Humanez J J Morales-Ruiz and J-A WeilldquoGaloisian approach to integrability of Schrodinger equationrdquoReports onMathematical Physics vol 67 no 3 pp 305ndash374 2011

[25] J J Kovacic ldquoAn algorithm for solving second order linearhomogeneous differential equationsrdquo Journal of Symbolic Com-putation vol 2 no 1 pp 3ndash43 1986

[26] M Baranger and K T Davies ldquoPeriodic trajectories for a two-dimensional nonintegrableHamiltonianrdquoAnnals of Physics vol177 no 2 pp 330ndash358 1987

[27] M Baranger M R Haggerty B Lauritzen D C Meredith andD Provost ldquoPeriodic orbits of nonscaling Hamiltonian systemsfrom quantum mechanicsrdquo Chaos An Interdisciplinary Journalof Nonlinear Science vol 5 no 1 pp 261ndash270 1995

[28] M Baranger K T Davies and J H Mahoney ldquoThe calculationof periodic trajectoriesrdquoAnnals of Physics vol 186 no 1 pp 95ndash110 1988

[29] S D Prado M A M De Aguiar J P Keating and R E DeCarvalho ldquoSemiclassical theory of magnetization for a two-dimensional non-interacting electron gasrdquo Journal of Physics AGeneral Physics vol 27 no 18 pp 6091ndash6106 1994

[30] C P Malta M A M De Aguiar and A M O De AlmeidaldquoQuantum signature of a period-doubling bifurcation and scarsof periodic orbitsrdquo Physical Review A Atomic Molecular andOptical Physics vol 47 no 3 pp 1625ndash1632 1993

[31] D Provost and M Baranger ldquoSemiclassical calculation of scarsfor a smooth potentialrdquo Physical Review Letters vol 71 no 5pp 662ndash665 1993

[32] P B Acosta-Humanez and D Blazquez-Sanz ldquoNon-integrability of some Hamiltonians with rational potentialsrdquoDiscrete and Continuous Dynamical Systems vol 10 no 2-3pp 265ndash293 2008

[33] G J Milburn J Corney E M Wright and D F WallsldquoQuantum dynamics of an atomic bose-einstein condensate ina double-well potentialrdquo Physical Review A Atomic Molecularand Optical Physics vol 55 no 6 pp 4318ndash4324 1997

[34] H G Roskos M C Nuss J Shah et al ldquoCoherentsubmillimeter-wave emission from charge oscillations ina double-well potentialrdquo Physical Review Letters vol 68 no 14pp 2216ndash2219 1992

[35] V Daniels M Vallieres and J-M Yuan ldquoChaotic scattering ona double well Periodic orbits symbolic dynamics and scalingrdquoChaos An Interdisciplinary Journal of Nonlinear Science vol 3no 4 pp 475ndash485 1993

[36] A JMaciejewski andM Przybylska ldquoNon-integrability of ABCflowrdquo Physics Letters A vol 303 no 4 pp 265ndash272 2002


[37] A JMaciejewskiM Przybylska L Simpson andW SzuminskildquoNon-integrability of the dumbbell and point mass problemrdquoCelestial Mechanics and Dynamical Astronomy vol 117 no 3pp 315ndash330 2013

[38] F E Lembarki and J Llibre ldquoPeriodic orbits of perturbed ellipticoscillators in 6d via averaging theoryrdquo Astrophysics and SpaceScience vol 361 no 10 pp 340ndash356 2016

[39] M Arribas A Elipe L Florıa and A Riaguas ldquoOscillators inresonance p q rrdquo Chaos Solitons amp Fractals vol 27 no 5 pp1220ndash1228 2006

[40] N D Caranicolas and K A Innanen ldquoPeriodic motion inperturbed elliptic oscillatorsrdquo The Astronomical Journal vol103 no 4 pp 1308ndash1312 1992

[41] A Elipe B Miller and M Vallejo ldquoBifurcations in a non-symmetric cubic potentialrdquo Astrophysics and Space Science vol300 pp 722ndash725 1995

[42] N D Caranicolas and E E Zotos ldquoInvestigating the natureof motion in 3d perturbed elliptic oscillators displaying exactperiodic orbitsrdquo Nonlinear Dynamics vol 69 no 4 pp 1795ndash1805 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


two degrees of freedom such as Toda lattice Hamiltonian[7] Kepler problems [8] Lie-Poisson system [9] and Euler-Poisson system [6]

An important progress was made by Morales RamisSimo [10 11] and Baider Churchill Rod and Singer [12] inthe end of 20th centuryThey replaced themonodromy groupof NVEwith the differential Galois group of NVE and showedthat the Liouville integrability of the nonlinear Hamiltoniansystem implies the integrability of the linear system NVE inthe sense of expressibility of solutions in closed form

Consider an analytic Hamiltonian system 119883119867 on a com-plex analytic symplectic manifold119872 and the equation of themotion for119867 reads119889119889119905119911 = 119883119867 119911 = (1199111 119911119899) isin C2119899 119905 isin C (1)

The variational equation (VE) along a particular solution 119911 =120601(119905) of (1) has the form of119889119889119905120585 = 119860 (119905) 120585 119860 (119905) = 120597119883119867120597119911 (120601 (119905)) 120585 isin C2119899 (2)

By using the linear first integral 119889119867(120601(119905)) of VE(2) (2) can bereduced into the NVE119889119889119905120578 = 119861 (119905) 120578 120578 isin C2(119899minus1) (3)

in suitable coordinates [10]

Theorem 1 (see [10]) Let 119867 be a Hamiltonian in C2119899 andlet Γ be a particular solution Assume that there exist 119899rational first integrals of 119883119867 (1) which are in involution andfunctionally independent in a neighborhood of Γ Then theidentity component of the differential Galois group of NVE(3)is Abelian

Remark 2 As was pointed in Ref [10] if the variational equa-tion is Fuchsian ie means that all singularities includinginfinity are regular one can extend the class of first integralsand get the meromorphic version of Theorem 1

Compared with Ziglinrsquos theorem the Morales-Ramistheory is more effective to study the nonintegrability ofthe Hamiltonian systems On the one hand the differentialGalois group is bigger than the monodromy group andthere are linear differential equationswith trivialmonodromygroup but with nontrivial differential Galois group On theother hand the differential Galois group is an algebraicgroup in particular a Lie group and can be calculated byinfinitesimal methods Since then Morales-Ramis theory hasbeen applied successfully for studying the nonintegrabilityof large numbers of physical models such as the planarthree-body problem [13ndash17] Hillrsquos problem [18] generalizedYang-Mills Hamiltonian [19] Wilberforce spring-pendulumproblem [20] and double pendula problem [21] It shouldbe pointed out that the differential Galois group can also beused to investigate the nonintegrability of general dynamicalsystems which may be non-Hamiltonian [22ndash24]

The aim of this paper is to investigate some Hamil-tonian systems including Nelson Hamiltonian a double-well potential Hamiltonian and perturbed elliptic oscillators

Hamiltonian By using Morales-Ramis theory and Kovacicrsquosalgorithm we will contribute to the understanding of thecomplexity or the topological structure of these systems fromthe point of view of nonintegrability

In order to apply theMorales-Ramis theory it is necessaryto know how to check if the identity component of thedifferential Galois group of a given linear system is AbelianGenerally speaking this is a hard work However effectivemethods to investigate this question for lower dimensionalsystems with rational coefficients do exist In particularfor second order equations over C(119909) the algorithm ofKovacic [25] allows to decidewhether the identity componentof the differential Galois group is solvable Based on theconsiderations above we can solve the problems in this paperaccording to the following steps

Firstly finding a nonconstant particular solution Γ ofthe considered Hamiltonian system Secondly calculatingthe variational equation along Γ furthermore we can getthe normal variational equation Then we transform thenormal variational equation to an equation with rationalcoefficients by using a transformation which does not changethe identity component of the differential Galois groupFinally we investigate the differential Galois group of theequation with rational coefficients with the help of Kovacicrsquosalgorithm

2 Main Results

21 Nonintegrability of Nelson Hamiltonian Consider theHamiltonian119867 = 12 (1199012119909 + 1199012119910) + (119910 minus 121199092)2 + 12058311990922 (4)

where 119909 119910 isin C 120583 is an arbitrary real parameter Its associatedHamiltonian system is = 119901119909119910 = 119901119910119909 = 119909 (2119910 minus 1199092 minus 120583) 119910 = 1199092 minus 2119910

(5)

This differential system is known as the Hamiltonian systemwith Nelson potential Interest in it is coming from nuclearphysics where the deep valley could represent a collectivedegree of freedom which remains coupled to other types ofexcitation [26]

The Nelson Hamiltonian resembles the Henon-Heilespotential and it also has a rich well-known periodic orbitstructure [27] Furthermore the motion which it generatescan be shown to be bounded and has been widely studied interms of periodic orbits [28] bifurcations [29] and quantummechanically [30 31]

The main goal of this section is to study the noninte-grability of system (5) by using Morales-Ramis theory andKovacicrsquos algorithm Our result is formulated in the followingtheorem




N = (119909 119910 119901119909 119901119910) isin C4 | 119909 = 119901119909 = 0 (6)


(7)



1205851120585212057811205782) (8)





Putting 1205851 = 119883 exp 12 int119886 (120591) 119889120591 (13)










0

H=005


minus04

minus03

minus02

minus01

01

02

03

04

px

H=001


minus02

minus015

minus01

minus005

0

005

01

015

px

H=015

minus08

minus06

minus04

minus02

0

02

04

06

px


0

H=01

minus05

minus04

minus03

minus02

minus01

01

02

03

04

05

px



(18)







M = (119909 119910 119901119909 119901119910) isin C4 | 119909 = 119901119909 = 0 (19)




( 1205851120585212057811205782)=( 0 0 1 00 0 0 1minus21205721205731199102 exp (minus1205731199102) 0 0 00 119872 0 0)(

1205851120585212057811205782)(21)




and 119886 (119911) = 120573119911 minus 22119911 119887 (119911) = minus 1205734119911 (25)


Then (24) becomes 11988310158401015840 = 119903 (119911)119883 (27)






and


(31)







(33)





(34)


120585112058521205853120578112057821205783)))))))

=((((((


120585112058521205853120578112057821205783))))))(35)



1205851120585212057811205782)(36)



1205851120578112058521205782)(37)














Appendix















119894

120572119894(119909 minus 119888119894)2 +sum119895 120573119895119909 minus 119889119895 (A5)


119894120572119894 +sum119895120573119895119889119895 (A6)



























Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018











N = (119909 119910 119901119909 119901119910) isin C4 | 119909 = 119901119909 = 0 (6)


(7)



1205851120585212057811205782) (8)





Putting 1205851 = 119883 exp 12 int119886 (120591) 119889120591 (13)










0

H=005


minus04

minus03

minus02

minus01

01

02

03

04

px

H=001


minus02

minus015

minus01

minus005

0

005

01

015

px

H=015

minus08

minus06

minus04

minus02

0

02

04

06

px


0

H=01

minus05

minus04

minus03

minus02

minus01

01

02

03

04

05

px



(18)







M = (119909 119910 119901119909 119901119910) isin C4 | 119909 = 119901119909 = 0 (19)




( 1205851120585212057811205782)=( 0 0 1 00 0 0 1minus21205721205731199102 exp (minus1205731199102) 0 0 00 119872 0 0)(

1205851120585212057811205782)(21)




and 119886 (119911) = 120573119911 minus 22119911 119887 (119911) = minus 1205734119911 (25)


Then (24) becomes 11988310158401015840 = 119903 (119911)119883 (27)






and


(31)







(33)





(34)


120585112058521205853120578112057821205783)))))))

=((((((


120585112058521205853120578112057821205783))))))(35)



1205851120585212057811205782)(36)



1205851120578112058521205782)(37)














Appendix















119894

120572119894(119909 minus 119888119894)2 +sum119895 120573119895119909 minus 119889119895 (A5)


119894120572119894 +sum119895120573119895119889119895 (A6)



























Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018









0

H=005


minus04

minus03

minus02

minus01

01

02

03

04

px

H=001


minus02

minus015

minus01

minus005

0

005

01

015

px

H=015

minus08

minus06

minus04

minus02

0

02

04

06

px


0

H=01

minus05

minus04

minus03

minus02

minus01

01

02

03

04

05

px



(18)







M = (119909 119910 119901119909 119901119910) isin C4 | 119909 = 119901119909 = 0 (19)




( 1205851120585212057811205782)=( 0 0 1 00 0 0 1minus21205721205731199102 exp (minus1205731199102) 0 0 00 119872 0 0)(

1205851120585212057811205782)(21)




and 119886 (119911) = 120573119911 minus 22119911 119887 (119911) = minus 1205734119911 (25)


Then (24) becomes 11988310158401015840 = 119903 (119911)119883 (27)






and


(31)







(33)





(34)


120585112058521205853120578112057821205783)))))))

=((((((


120585112058521205853120578112057821205783))))))(35)



1205851120585212057811205782)(36)



1205851120578112058521205782)(37)














Appendix















119894

120572119894(119909 minus 119888119894)2 +sum119895 120573119895119909 minus 119889119895 (A5)


119894120572119894 +sum119895120573119895119889119895 (A6)



























Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018










( 1205851120585212057811205782)=( 0 0 1 00 0 0 1minus21205721205731199102 exp (minus1205731199102) 0 0 00 119872 0 0)(

1205851120585212057811205782)(21)




and 119886 (119911) = 120573119911 minus 22119911 119887 (119911) = minus 1205734119911 (25)


Then (24) becomes 11988310158401015840 = 119903 (119911)119883 (27)






and


(31)







(33)





(34)


120585112058521205853120578112057821205783)))))))

=((((((


120585112058521205853120578112057821205783))))))(35)



1205851120585212057811205782)(36)



1205851120578112058521205782)(37)














Appendix















119894

120572119894(119909 minus 119888119894)2 +sum119895 120573119895119909 minus 119889119895 (A5)


119894120572119894 +sum119895120573119895119889119895 (A6)



























Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018










(33)





(34)


120585112058521205853120578112057821205783)))))))

=((((((


120585112058521205853120578112057821205783))))))(35)



1205851120585212057811205782)(36)



1205851120578112058521205782)(37)














Appendix















119894

120572119894(119909 minus 119888119894)2 +sum119895 120573119895119909 minus 119889119895 (A5)


119894120572119894 +sum119895120573119895119889119895 (A6)



























Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018
















Appendix















119894

120572119894(119909 minus 119888119894)2 +sum119895 120573119895119909 minus 119889119895 (A5)


119894120572119894 +sum119895120573119895119889119895 (A6)



























Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018































Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018


























































Journal of












Journal of







Volume 2018






Volume 2018








Complex Dynamics of Some Hamiltonian Systems...

Documents

Transcript of Complex Dynamics of Some Hamiltonian Systems...