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 SIGMA, 2019, Volume 15, 049, 17 pp. (Mi sigma1485)

Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

Kazuyuki Yagasaki, Shogo Yamanaka

Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan

Abstract: We consider a class of two-degree-of-freedom Hamiltonian systems with saddle-centers connected by heteroclinic orbits and discuss some relationships between the existence of transverse heteroclinic orbits and nonintegrability. By the Lyapunov center theorem there is a family of periodic orbits near each of the saddle-centers, and the Hessian matrices of the Hamiltonian at the two saddle-centers are assumed to have the same number of positive eigenvalues. We show that if the associated Jacobian matrices have the same pair of purely imaginary eigenvalues, then the stable and unstable manifolds of the periodic orbits intersect transversely on the same Hamiltonian energy surface when sufficient conditions obtained in previous work for real-meromorphic nonintegrability of the Hamiltonian systems hold; if not, then these manifolds intersect transversely on the same energy surface, have quadratic tangencies or do not intersect whether the sufficient conditions hold or not. Our theory is illustrated for a system with quartic single-well potential and some numerical results are given to support the theoretical results.

Keywords: nonintegrability, Hamiltonian system, heteroclinic orbits, saddle-center, Melnikov method, Morales–Ramis theory, differential Galois theory, monodromy.

 Funding Agency Grant Number Japan Society for the Promotion of Science JP17H02859JP17J01421 This work was partially supported by Japan Society for the Promotion of Science, Kekenhi Grant Numbers JP17H02859 and JP17J01421.

DOI: https://doi.org/10.3842/SIGMA.2019.049

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Full text: https://www.imath.kiev.ua/~sigma/2019/049/
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ArXiv: 1906.?????
MSC: 37J30, 34C28, 37C29
Received: January 29, 2019; in final form June 21, 2019; Published online July 2, 2019
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Citation: Kazuyuki Yagasaki, Shogo Yamanaka, “Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers”, SIGMA, 15 (2019), 049, 17 pp.

Citation in format AMSBIB
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