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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the multiplicity of periodic orbits and homoclinics near critical energy levels of Hamiltonian systems in $ \mathbb{R}^4$


Authors: Naiara V. de Paulo and Pedro A. S. Salomão
Journal: Trans. Amer. Math. Soc. 372 (2019), 859-887
MSC (2010): Primary 37J05; Secondary 37J45, 53D35
DOI: https://doi.org/10.1090/tran/7568
Published electronically: April 18, 2019
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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.


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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
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

DOI: https://doi.org/10.1090/tran/7568
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.
Article copyright: © Copyright 2019 American Mathematical Society