Complex Dynamics of Some Hamiltonian Systems: Nonintegrability of Equations of Motion

Qu, Jingjia

doi:https://doi.org/10.1155/2019/9326947

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2019 | Article ID 9326947 | https://doi.org/10.1155/2019/9326947

Complex Dynamics of Some Hamiltonian Systems: Nonintegrability of Equations of Motion

Jingjia Qu^1,2

Academic Editor: Sandro Wimberger

Received06 Nov 2018

Revised23 Jan 2019

Accepted11 Feb 2019

Published03 Mar 2019

Abstract

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

1. Introduction

The topic of integrability and nonintegrability for systems of nonlinear differential equations is one of the central problems in the qualitative theory of ODEs. A system can be regarded as integrable if it admits such a number of first integrals or other tensor invariants that it is solvable by quadratures. The existence of functionally independent first integrals can help us reduce the study of the considered -dimensional system to -dimensional one. Restricted on the level sets of first integrals, we can understand the complexity and topological structure of the considered dynamical system. For example, if a Hamiltonian system with degrees of freedom is integrable in the sense of Liouville, i.e., it has functionally independent first integrals in involution, then the invariant sets associated with first integrals are generically diffeomorphic to tori, cylinders, or planes inside the phase space. If the system is not integrable, we can expect it may exhibit a variety of chaotic phenomena or complex dynamical behavior.

In general, it is difficult to determine whether a system is integrable or not; see [1, 2] for instance. For all we know, the first results on the nonintegrability should go back to Poincaré [3], who found the influence of resonances of the eigenvalues of the monodromy matrix associated to a periodic orbit for small generic perturbations of integrable systems. Then, in the study of the rigid body with one fixed point [4], Kovalevskaya showed that a given system whose singularities are only poles turns out to be integrable in some sense. The idea of Kovalevskaya was the first trying to detect the integrability of dynamical systems by studying the singular structure of their general solutions.

Ziglin [5, 6] investigated the existence of meromorphic first integrals for analytic Hamiltonian systems. Based on properties of the monodromy group of a normal variational equation (NVE) along a complex integral curve, he presented necessary conditions for a -dimensional complex analytic Hamiltonian system to have functionally independent rational first integrals. It should be pointed out that Ziglin’s theorem does not deal with the integrability in the Liouville sense of the Hamiltonian system with arbitrary number of degrees of freedom in his original formulation since the considered first integrals may not be in involution. However, since the Poisson bracket of first integrals and the Hamiltonian always vanishes, Ziglin’s results can be applied to study nonintegrability of the Hamiltonian system with 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, Ramis, Simó [10, 11], and Baider, Churchill, Rod, and Singer [12] in the end of 20th century. They replaced the monodromy group of NVE with the differential Galois group of NVE and showed that the Liouville integrability of the nonlinear Hamiltonian system implies the integrability of the linear system NVE in the sense of expressibility of solutions in closed form.

Consider an analytic Hamiltonian system on a complex analytic symplectic manifold , and the equation of the motion for readsThe variational equation (VE) along a particular solution of (1) has the form ofBy using the linear first integral of VE(2), (2) can be reduced into the NVEin suitable coordinates [10].

Theorem 1 (see [10]). Let be a Hamiltonian in , and let be a particular solution. Assume that there exist rational first integrals of (1) which are in involution and functionally independent in a neighborhood of . Then, the identity component of the differential Galois group of NVE(3) is Abelian.

Remark 2. As was pointed in Ref. [10], if the variational equation is Fuchsian, i.e., means that all singularities including infinity are regular, one can extend the class of first integrals and get the meromorphic version of Theorem 1.

Compared with Ziglin’s theorem, the Morales-Ramis theory is more effective to study the nonintegrability of the Hamiltonian systems. On the one hand, the differential Galois group is bigger than the monodromy group and there are linear differential equations with trivial monodromy group but with nontrivial differential Galois group. On the other hand, the differential Galois group is an algebraic group, in particular, a Lie group, and can be calculated by infinitesimal methods. Since then, Morales-Ramis theory has been applied successfully for studying the nonintegrability of large numbers of physical models, such as the planar three-body problem [13–17], Hill’s problem [18], generalized Yang-Mills Hamiltonian [19], Wilberforce spring-pendulum problem [20], and double pendula problem [21]. It should be pointed out that the differential Galois group can also be used to investigate the nonintegrability of general dynamical systems which may be non-Hamiltonian [22–24].

The aim of this paper is to investigate some Hamiltonian systems, including Nelson Hamiltonian, a double-well potential Hamiltonian, and perturbed elliptic oscillators Hamiltonian. By using Morales-Ramis theory and Kovacic’s algorithm, we will contribute to the understanding of the complexity or the topological structure of these systems from the point of view of nonintegrability.

In order to apply the Morales-Ramis theory, it is necessary to know how to check if the identity component of the differential Galois group of a given linear system is Abelian. Generally speaking, this is a hard work. However, effective methods to investigate this question for lower dimensional systems with rational coefficients do exist. In particular, for second order equations over the algorithm of Kovacic [25] allows to decide whether the identity component of the differential Galois group is solvable. Based on the considerations above, we can solve the problems in this paper according to the following steps.

Firstly, finding a nonconstant particular solution of the considered Hamiltonian system. Secondly, calculating the variational equation along , furthermore, we can get the normal variational equation. Then, we transform the normal variational equation to an equation with rational coefficients by using a transformation which does not change the identity component of the differential Galois group. Finally, we investigate the differential Galois group of the equation with rational coefficients with the help of Kovacic’s algorithm.

2. Main Results

2.1. Nonintegrability of Nelson Hamiltonian

Consider the Hamiltonianwhere , , is an arbitrary real parameter. Its associated Hamiltonian system isThis differential system is known as the Hamiltonian system with Nelson potential. Interest in it is coming from nuclear physics, where the deep valley could represent a collective degree of freedom which remains coupled to other types of excitation [26].

The Nelson Hamiltonian resembles the Hénon-Heiles potential and it also has a rich well-known periodic orbit structure [27]. Furthermore, the motion which it generates can be shown to be bounded and has been widely studied, in terms of periodic orbits [28], bifurcations [29], and quantum mechanically [30, 31].

The main goal of this section is to study the nonintegrability of system (5) by using Morales-Ramis theory and Kovacic’s algorithm. Our result is formulated in the following theorem.

Theorem 3. The system (5) is nonintegrable in the sense of Liouville.

Proof of Theorem 3. It is easy to find that the manifoldis invariant with respect to the flow of (5), and system (5) has a solutionwhich lies in . Let be variations in . Then, the variational equation of (4) along readsSince the particular solution lies on , the equation for variables forms a closed subsystem which is the so-called normal variational equation and is equivalent toMaking a change of variable , we can transform (10) intoThe prime denotes the derivative with respect to , wherePuttingthen (11) can be changed to the reduced formwhereIn what follows, we will show that the differential Galois group of (14) is . Clearly, the set of singularities of equation (14) is , the order at and is two, and the order at is one; necessary conditions for Cases 1 and 3 are not satisfied and only Case 2 should be analysed; see Theorem A.2 in Appendix. Indeed, by simple computation, we getHowever, for any , , , , so Case 2 in Theorem A.2 in Appendix cannot hold, too. By Theorem A.2 in Appendix, the differential Galois group of (14) is in Case 4, so the corresponding identity component of (14) is also .
Assume system (5) is integrable in the sense of Liouville, then by Theorem 1 we conclude that the identity component of the differential Galois group of (10) is Abelian. Note that the transformation does not change the identity component of this group (page 31, Theorem 2.5 [10]). Thus the identity component of (11) is Abelian. Since an Abelian group is always a solvable group and the solvability of (11) coincides with that of (14), we show that the identity component of (14) is solvable. However, is not solvable. We can complete the proof of this theorem.

We remark that (10) is a particular case of Mathieu Equation and its solvability has been studied by P. Acosta-Humánez et al. [32]. In their works, they also investigated a large class of Hamiltonians with rational potentials which is close to Nelson potential.

By using Symplectic algorithm and Matlab one can observe a rich variety of behavior of the Nelson system in Poincaré cross section as the energy increases. For example, we consider the Poincaré cross section with ; Figure 1 shows that the section changes from regular pattern to irregular pattern as the energy level increases from 0.01 to 0.15; this indeed implies the phenomena of chaos. Theorem 3 shows that (5) is nonintegrable. We all know that lack of first integrals can imply a complex behavior of phase curves of the system, and the numerical result just illustrates this point.

2.2. Nonintegrability of a Double-Well Potential Hamiltonian

Next, we consider the planar Hamiltonian with a double-well potentialwhere are real parameters. Its associated Hamiltonian equation isThe motivation for the choice of the potential comes from the interest of this potential in chaotic scattering [33, 34]. This system can be considered as a model to describe the electron scattering on the ion or the prototype for the trapping of incoming stellar objects by a collection of massive objects such as a solar system [35].

Numerical tests in [35] show that the motion of system (18) has sensitive dependence on initial conditions and chaotic scattering occurs when . However, such tests do not exclude the possibility that the system (18) admits complex or chaotic behavior for other values of . In this work, from a view of the integrability, we study the nonintegrability of system (18) and contribute to understanding the complexity of system (18).

In the case of , obviously, the Hamiltonian system (18) is integrable. Suppose ; we proceed to prove the theorem below.

Theorem 4. For , the Hamiltonian system (18) is nonintegrable in the sense of Liouville.

Proof of Theorem 4. It is easy to see that the manifoldis invariant with respect to the flow generated by system (18). Indeed, system (18) restricted to is given bywhich is completely integrable with the first integral . Fixing the level set , we obtain a particular solution with and denote by the phase curve corresponding to this solution. The variational equation of (18) along the solution is given bywhere . The corresponding normal variational equation reads or equivalently,The right-hand side of this equation is nontrivial provided . Making a change of variable , then (23) becomesandFurthermore, letThen (24) becomeswhereThen we show that the differential Galois group of (27) is . It follows from Theorem A.1 in Appendix that we need only to check that Cases 1, 2, and 3 cannot hold. Due to , we see that the function admits two singular points with order two, and with order zero. We can easily know that Case 3 in Theorem A.2 cannot hold. Moreover, the difference of exponents at is zero. Then in a neighbourhood of , two independent local solutions have the form where and are analytic at and . It follows that the differential Galois group can be only full triangular of (for details see [36, 37]). Hence, only Case 1 or 4 of the Kovacic’s algorithm is possible.
For Case 1, a simple computation leads toandIt is not difficult to check that, for any , is not a nonnegative integer. Therefore, Case 1 cannot hold.
Based on above discussion, we can make the conclusion that Case 4 in Theorem A.1 in Appendix holds; namely, the differential Galois group of (27) is . Using the same argument as in the proof of Theorem 3, we can easily complete the proof of this theorem.

2.3. Nonintegrability of Perturbed Elliptic Oscillators Hamiltonian

We study the following Hamiltonian which appeared in [38]:where is an arbitrary real parameter. The Hamiltonian we consider above consists of three coupled harmonic oscillators known as perturbed elliptic oscillators. Its associated Hamiltonian system isThe reason for the choice of this Hamiltonian is that perturbed elliptic oscillators appear very often in galactic dynamics and atomic physics [39–41]. Perturbed elliptic oscillators display exact periodic orbits, interesting sticky orbits together with large chaotic regions [42]. We want to show that the system there may cause complex behavior from the view of nonintegrability.

If , it is easy to get the conclusion that the Hamiltonian system (33) is integrable. If , we obtain the following theorem.

Theorem 5. Assume ; then the system (33) is nonintegrable in the sense of Liouville.

Proof of Theorem 5. Obviously, system (33) has a periodic solutionIt is easy to check that the variational equation of (32) along isand corresponding normal variational equation isMaking a change of the location of the variable, we get the following equation:Clearly, the normal variational equation (37) along is a system of two uncoupled second order linear differential equationsWe denote by , , and the differential Galois group of the equations (38), (39), and (37), respectively, over the field of rational functions on . As a representation of an element of is of the formwhere is the null matrix, , and , the identity component of is not Abelian if the identity component of or is not Abelian. Then, we turn to consider the normal variational equation (38) and analyze the differential Galois group .
In order to do that, we make the usual change of variable ; then (38) becomeswhereIntroducing the change of variablewe have that the equation (41) becomeswhereThis theorem will follow from Theorem 1, if we can show that the differential Galois group of (44) is .
Note that the set of singularities of equation (44) is . The order at and is 2. It follows from that the order at is 1. We now proceed as in the proof of Theorem 3. By Theorem A.2 in Appendix Cases 1 and 3 do not occur.
For Case 2, by simple computation, we getHowever ; it is not a nonnegative integer, so Case 2 cannot happen. By Theorem A.2 in Appendix, we know that Cases 1, 2, and 3 in Theorem A.1 in Appendix cannot hold; we have thus proved the theorem.

Appendix

A. Kovacic’s Algorithm

Consider the differential equationwhere is the field of rational functions defined on the complex plane .

We recall that any general second order linear differential equation can be transformed to the form (A.1) by change of variable. Indeed, for equationif we make the change of variable , then (A.2) becomes

To avoid triviality, we assume that is not a constant. Then, one has the following.

Theorem A.1 (see [25]). There are four cases that can occur.
Case 1. is conjugate to a triangular group. Then (A.1) has a solution of the form with .
Case 2. is not of Case 1, but it is conjugate to a subgroup ofThen (A.1) has a solution of the form with algebraic over of degree 2.
Case 3. is not of Cases 1 and 2, but it is a finite group. Then all solutions of (A.1) are algebraic over .
Case 4. . Then (A.1) is not integrable in Liouville sense.

Furthermore, corresponding to the four cases listed in Theorem A.1, there are some easy necessary conditions for Cases 1, 2, and 3 which consequently form a sufficient condition for Case 4. Since , it can be represented as with relatively prime. Then the poles of are the zeros of and we mean the order of the pole by the multiplicity of the zero of . By the order of at we shall refer the order of as a zero of , i.e., . Then we have the following.

Theorem A.2 (see [25]). The following conditions are necessary for the respective cases of Theorem A.1 to hold.
Case 1. Every pole of must have even order or else have order 1. The order of at must be even or else be greater than 2.
Case 2. must have at least one pole that either has odd order greater than 2 or else has order 2.
Case 3. The order of a pole of cannot exceed 2 and the order of at must be at least 2. If the partial fraction expansion of isthen , for each , and . Furthermore, if we letthen .

In what follows, we describe the Kovacic’s algorithm [25]. Let be the set of singularities of linear differential equation (A.1) containing the poles of and .

A.1. Case 1 of Kovacic’s Algorithm

Step 1. For each we define , as follows.
(a) If is a pole of order , then , .
(b) If is a pole of order , then , , where is coefficients of in the partial fraction expansion for .
(c) If is a pole of order , then of negative order part of the Laurent series expansion of at , , where is the coefficient of in minus the coefficient of in .
(d) If the order of at is , then , , .
(e) If the order of at is , then , , where is coefficients of in the Laurent series expansion of at .
(f) If is a pole of order , then of positive order part of the Laurent series expansion of at , , where is the coefficient of in minus the coefficient of in .

Step 2. Let , where for any , and for any . If is a nonnegative integer, then let ; otherwise, the family is discarded. If no families remain under consideration, Case 1 of Theorem A.1 cannot happen.

Step 3. For each family retained from Step 2, we search for a monic polynomial of degree such that the equation holds. If such a polynomial exists, then is a solution of . If no such polynomial is found for any family retained from Step 2, Case 1 of Theorem A.1 cannot happen.

A.2. Case 2 of Kovacic’s Algorithm

Step 1. For each we define as follows.
(a) If is a pole of order , then .
(b) If is a pole of order , then , where is coefficients of in the partial fraction expansion for .
(c) If is a pole of order , then .
(d) If the order of at is , then .
(e) If the order of at is , then , where is coefficients of in the Laurent series expansion of at .
(f) If is a pole of order , then .

Step 2. Let , where for any , and for any . If is a nonnegative integer, then let ; otherwise, the family is discarded. If no families remain under consideration, Case 2 of Theorem A.1 cannot happen.

Step 3. For each family retained from Step 2, we search for a monic polynomial of degree such that the equation holds. If such a polynomial exists, let and let be a solution of the equation ; then is a solution of . If no such polynomial is found for any family retained from Step 2, Case 2 of Theorem A.1 cannot happen.

A.3. Case 3 of Kovacic’s Algorithm

Step 1. For fixed, we define with as follows.
(a) If is a pole of order , then .
(b) If is a pole of order , then , where is coefficients of in the partial fraction expansion for .
(c) , where is coefficients of in the Laurent series expansion of at .

Step 2. Let , where for any , and for any . If is a nonnegative integer, then let , ; otherwise, the family is discarded. If no families remain under consideration, Case 3 of Theorem A.1 cannot happen.

Step 3. For each family retained from Step 2, we search for a monic polynomial of degree such that the recursive equations with hold. If such a polynomial exists, let be a solution of the equation ; then is a solution of . If no such polynomial is found for any family retained from Step 2, Case 3 of Theorem A.1 cannot happen.

Data Availability

All data included in this study are available upon request by contact 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

J. J. Morales-Ruiz, “Meromorphic nonintegrability of Hamiltonian systems,” Reports on Mathematical Physics, vol. 48, no. 1-2, pp. 183–194, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
K. Goździewski, A. J. Maciejewski, and Z. Niedzielska, “Searching for integrable systems. An application of Kozlov theorem,” Reports on Mathematical Physics, vol. 46, no. 1-2, pp. 175–182, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
V. V. Kozlov, “Integrability and non-integrability in Hamiltonian mechanics,” Russian Mathematical Surveys, vol. 38, no. 1, pp. 1–76, 1983.
View at: Publisher Site | Google Scholar
S. Kowalevski, “Sur le problème de la rotation d’un corps solide autour d’un point fixe,” Acta Mathematica, vol. 12, no. 1, pp. 177–232, 1889.
View at: Publisher Site | Google Scholar | MathSciNet
S. L. Ziglin, “Branching of solutions and the nonexistence of first integrals in Hamiltonian mechanics.I,” Functional Analysis and Its Applications, vol. 16, pp. 181–189, 1982.
View at: Google Scholar | MathSciNet
S. L. Ziglin, “Branching of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II,” Functional Analysis and its Applications, vol. 17, no. 1, pp. 6–17, 1983.
View at: Publisher Site | Google Scholar
H. Yoshida, “Nonintegrability of the truncated Toda lattice Hamiltonian at any order,” Communications in Mathematical Physics, vol. 116, no. 4, pp. 529–538, 1988.
View at: Publisher Site | Google Scholar | MathSciNet
H. Yoshida, “Nonintegrability of a class of perturbed Kepler problems in two dimensions,” Physics Letters A, vol. 120, no. 8, pp. 388–390, 1987.
View at: Publisher Site | Google Scholar | MathSciNet
A. J. Maciejewski and K. Goździewski, “Numerical evidence of nonintegrability of certain lie-poisson system,” Reports on Mathematical Physics, vol. 44, no. 1-2, pp. 133–142, 1999.
View at: Publisher Site | Google Scholar | MathSciNet
J. Morales-Ruiz, Differential Galois Theory and Nonintegrability of Hamiltonian Systems, vol. 179 of Progress in Mathematics, Birkhäuser, 1999.
View at: MathSciNet
J. Morales-Ruiz and C. Simó, “Picard-vessiot theory and Ziglin’s theorem,” Journal of Differential Equations, vol. 107, no. 1, pp. 140–162, 1994.
View at: Publisher Site | Google Scholar | MathSciNet
A. Baider, R. C. Churchill, D. L. Rod, and M. Singer, “On the infinitesimal geometry of integrable systems,” in Mechanics Day (Waterloo, ON, 1992), vol. 7 of Fields Institute Communications, vol. 7, pp. 5–56, American Mathematical Society, Providence, RI, USA, 1996.
View at: Google Scholar | MathSciNet
A. V. Tsygvintsev, “The meromorphic non-integrability of the three-body problem,” Journal für die Reine und Angewandte Mathematik, vol. 537, pp. 127–149, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
D. Boucher and J.-A. Weil, “Application of J.-J. Morales and J.-P. Ramis’ theorem to test the non-complete integrability of the planar three-body problem,” in From Combinatorics to Dynamical Systems, C. Mitschi and F. Fauvet, Eds., vol. 3 of IRMA Lectures in Mathematics and Theoretical Physics, pp. 163–177, Walter De Gruyter, Berlin, Germany, 2003.
View at: Google Scholar | MathSciNet
A. V. Tsygvintsev, “Non-existence of new meromorphic first integrals in the planar three-body problem,” Celestial Mechanics and Dynamical Astronomy, vol. 86, no. 3, pp. 237–247, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
P. B. Acosta-Humánez, M. Álvarez-Ramírez, and J. Delgado, “Non-integrability of some few body problems in two degrees of freedom,” Qualitative Theory of Dynamical Systems, vol. 8, no. 2, pp. 209–239, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
A. J. Maciejewski and M. Przybylska, “Non-integrability of the three-body problem,” Celestial Mechanics and Dynamical Astronomy, vol. 110, no. 1, pp. 17–30, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
J. J. Morales-Ruiz, C. Simó, and S. Simon, “Algebraic proof of the non-integrability of Hill’s problem,” Ergodic Theory and Dynamical Systems, vol. 25, no. 4, pp. 1237–1256, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
S. Shi and W. Li, “Non-integrability of generalized Yang-Mills Hamiltonian system,” Discrete and Continuous Dynamical Systems, vol. 33, no. 4, pp. 1645–1655, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
P. B. Acosta-Humánez, M. Álvarez-Ramírez, D. Blázquez-Sanz, and J. Delgado, “Non-integrability criterium for normal variational equations around an integrable subsystem and an example: the wilberforce spring-pendulum,” Discrete and Continuous Dynamical Systems, vol. 33, no. 3, pp. 965–986, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
T. Stachowiak and W. Szumiński, “Non-integrability of restricted double pendula,” Physics Letters A, vol. 379, no. 47-48, pp. 3017–3024, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
W. Li and S. Shi, “Galoisian obstruction to the integrability of general dynamical systems,” Journal of Differential Equations, vol. 252, no. 10, pp. 5518–5534, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
M. Ayoul and N. T. Zung, “Galoisian obstructions to non-Hamiltonian integrability,” Comptes Rendus Mathematique Academie des Sciences, Paris, vol. 348, no. 23-24, pp. 1323–1326, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
P. B. Acosta-Humánez, J. J. Morales-Ruiz, and J.-A. Weil, “Galoisian approach to integrability of Schrödinger equation,” Reports on Mathematical Physics, vol. 67, no. 3, pp. 305–374, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
J. J. Kovacic, “An algorithm for solving second order linear homogeneous differential equations,” Journal of Symbolic Computation, vol. 2, no. 1, pp. 3–43, 1986.
View at: Publisher Site | Google Scholar | MathSciNet
M. Baranger and K. T. Davies, “Periodic trajectories for a two-dimensional nonintegrable Hamiltonian,” Annals of Physics, vol. 177, no. 2, pp. 330–358, 1987.
View at: Publisher Site | Google Scholar | MathSciNet
M. Baranger, M. R. Haggerty, B. Lauritzen, D. C. Meredith, and D. Provost, “Periodic orbits of nonscaling Hamiltonian systems from quantum mechanics,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 5, no. 1, pp. 261–270, 1995.
View at: Publisher Site | Google Scholar | MathSciNet
M. Baranger, K. T. Davies, and J. H. Mahoney, “The calculation of periodic trajectories,” Annals of Physics, vol. 186, no. 1, pp. 95–110, 1988.
View at: Publisher Site | Google Scholar | MathSciNet
S. D. Prado, M. A. M. De Aguiar, J. P. Keating, and R. E. De Carvalho, “Semiclassical theory of magnetization for a two-dimensional non-interacting electron gas,” Journal of Physics A: General Physics, vol. 27, no. 18, pp. 6091–6106, 1994.
View at: Google Scholar
C. P. Malta, M. A. M. De Aguiar, and A. M. O. De Almeida, “Quantum signature of a period-doubling bifurcation and scars of periodic orbits,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 47, no. 3, pp. 1625–1632, 1993.
View at: Publisher Site | Google Scholar
D. Provost and M. Baranger, “Semiclassical calculation of scars for a smooth potential,” Physical Review Letters, vol. 71, no. 5, pp. 662–665, 1993.
View at: Publisher Site | Google Scholar
P. B. Acosta-Humánez and D. Blázquez-Sanz, “Non-integrability of some Hamiltonians with rational potentials,” Discrete and Continuous Dynamical Systems, vol. 10, no. 2-3, pp. 265–293, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic bose-einstein condensate in a double-well potential,” Physical Review A: Atomic, Molecular and Optical Physics, vol. 55, no. 6, pp. 4318–4324, 1997.
View at: Publisher Site | Google Scholar
H. G. Roskos, M. C. Nuss, J. Shah et al., “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Physical Review Letters, vol. 68, no. 14, pp. 2216–2219, 1992.
View at: Publisher Site | Google Scholar
V. Daniels, M. Vallieres, and J.-M. Yuan, “Chaotic scattering on a double well: Periodic orbits, symbolic dynamics, and scaling,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 3, no. 4, pp. 475–485, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
A. J. Maciejewski and M. Przybylska, “Non-integrability of ABC flow,” Physics Letters A, vol. 303, no. 4, pp. 265–272, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
A. J. Maciejewski, M. Przybylska, L. Simpson, and W. Szumiński, “Non-integrability of the dumbbell and point mass problem,” Celestial Mechanics and Dynamical Astronomy, vol. 117, no. 3, pp. 315–330, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
F. E. Lembarki and J. Llibre, “Periodic orbits of perturbed elliptic oscillators in 6d via averaging theory,” Astrophysics and Space Science, vol. 361, no. 10, pp. 340–356, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
M. Arribas, A. Elipe, L. Floría, and A. Riaguas, “Oscillators in resonance p: q: r,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1220–1228, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
N. D. Caranicolas and K. A. Innanen, “Periodic motion in perturbed elliptic oscillators,” The Astronomical Journal, vol. 103, no. 4, pp. 1308–1312, 1992.
View at: Publisher Site | Google Scholar
A. Elipe, B. Miller, and M. Vallejo, “Bifurcations in a non-symmetric cubic potential,” Astrophysics and Space Science, vol. 300, pp. 722–725, 1995.
View at: Google Scholar
N. D. Caranicolas and E. E. Zotos, “Investigating the nature of motion in 3d perturbed elliptic oscillators displaying exact periodic orbits,” Nonlinear Dynamics, vol. 69, no. 4, pp. 1795–1805, 2012.
View at: Publisher Site | Google Scholar | MathSciNet

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

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