| Full text | |
| Author(s): |
Total Authors: 2
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| Affiliation: | [1] Univ Fed Santa Catarina, Dept Matemat, Rua Joao Pessoa 2514, Blumenau, SC - Brazil
[2] Univ Sao Paulo, Dept Matemat, Inst Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil
Total Affiliations: 2
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| Document type: | Journal article |
| Source: | TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY; v. 372, n. 2, p. 859-887, JUL 15 2019. |
| Web of Science Citations: | 0 |
| Abstract | |
We study two-degree-of-freedom Hamiltonian systems. Let us assume that the zero energy level of a real-analytic Hamiltonian function H : R-4 -> R contains a saddle-center equilibrium point lying in a strictly convex sphere-like singular subset S-0 subset of H-1 (0). From previous work {[}Mem. Amer. Math. Soc. 252 (2018)] we know that for any small energy E > 0, the energy level H-1 (E) contains a closed 3-ball S-E in a neighborhood of S-0 admitting a singular foliation called 2 - 3 foliation. One of the binding orbits of this singular foliation is the Lyapunoff orbit P-2,P- E contained in the center manifold of the saddle-center. The other binding orbit lies in the interior of S-E and spans a one parameter family of disks transverse to the Hamiltonian vector field. In this article we show that the 2-3 foliation forces the existence of infinitely many periodic orbits and infinitely many homoclinics to P-2,P- E in S-E. Moreover, if the branches of the stable and unstable manifolds of P-2,P- E inside S-E do not coincide, then the Hamiltonian flow on S-E has positive topological entropy. We also present applications of these results to some classical Hamiltonian systems. (AU) | |
| FAPESP's process: | 16/25053-8 - Dynamics and geometry in low dimensions |
| Grantee: | André Salles de Carvalho |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 14/08113-1 - Systems of transverse sections in classical mechanics |
| Grantee: | Naiara Vergian de Paulo |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |