
SIGMA 15 (2019), 049, 17 pages arXiv:1907.01161
https://doi.org/10.3842/SIGMA.2019.049
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems
Heteroclinic Orbits and Nonintegrability in TwoDegreeofFreedom Hamiltonian Systems with SaddleCenters
Kazuyuki Yagasaki and Shogo Yamanaka
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, YoshidaHonmachi, Sakyoku, Kyoto 6068501, Japan
Received January 29, 2019, in final form June 21, 2019; Published online July 02, 2019
Abstract
We consider a class of twodegreeoffreedom Hamiltonian systems with saddlecenters 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 saddlecenters, and the Hessian matrices of the Hamiltonian at the two saddlecenters 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 realmeromorphic 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 singlewell potential and some numerical results are given to support the theoretical results.
Key words: nonintegrability; Hamiltonian system; heteroclinic orbits; saddlecenter; Melnikov method; MoralesRamis theory; differential Galois theory; monodromy.
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