On the multiplicity of periodic orbits and homoclinics near critical energy levels of Hamiltonian systems in $\mathbb {R}^4$
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- by Naiara V. de Paulo and Pedro A. S. Salomão PDF
- Trans. Amer. Math. Soc. 372 (2019), 859-887 Request permission
Abstract:
We study two-degree-of-freedom Hamiltonian systems. Let us assume that the zero energy level of a real-analytic Hamiltonian function $H:\mathbb {R}^4 \to \mathbb {R}$ contains a saddle-center equilibrium point lying in a strictly convex sphere-like singular subset $S_0\subset H^{-1}(0)$. 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 $P_{2,E}$ 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 $P_{2,E}$ in $S_E$. Moreover, if the branches of the stable and unstable manifolds of $P_{2,E}$ 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.References
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Additional Information
- Naiara V. de Paulo
- Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, Rua João Pessoa, 2514, Bairro Velha, Blumenau SC, Brazil 89036-004
- MR Author ID: 1260491
- Email: naiara.vergian@ufsc.br
- Pedro A. S. Salomão
- Affiliation: Instituto de Matemática e Estatística, Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo SP, Brazil 05508-090
- Email: psalomao@ime.usp.br
- Received by editor(s): December 13, 2017
- Received by editor(s) in revised form: February 12, 2018, and February 28, 2018
- Published electronically: April 18, 2019
- Additional Notes: The first author was partially supported by FAPESP 2014/08113-1.
The second author was partially supported by CNPq 306106/2016-7 and by FAPESP 2016/25053-8. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 859-887
- MSC (2010): Primary 37J05; Secondary 37J45, 53D35
- DOI: https://doi.org/10.1090/tran/7568
- MathSciNet review: 3968790