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Systems of Transversal Sections near Critical Energy Levels of Hamiltonian Systems in $\mathbb {R}^4$
About this Title
Naiara V. de Paulo, Universidade Federal de Santa Catarina–Departmento de Matemática, Rua João Pessoa, 2750–Bairro Velha–Blumenau SC, Brazil 89036-256 and Pedro A. S. Salomão, Universidade de São Paulo, Instituto de Matemática e Estatística – Departamento de Matemática, Rua do Matão, 1010 - Cidade Universitária - São Paulo SP, Brazil 05508-090
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 252, Number 1202
ISBNs: 978-1-4704-2801-3 (print); 978-1-4704-4373-3 (online)
DOI: https://doi.org/10.1090/memo/1202
Published electronically: January 25, 2018
Keywords: Hamiltonian dynamics,
systems of transversal sections,
pseudo-holomorphic curves,
periodic orbits,
homoclinic orbits
MSC: Primary 53D35; Secondary 37J55, 37J45
Table of Contents
Chapters
- 1. Introduction
- 2. Proof of the main statement
- 3. Proof of Proposition
- 4. Proof of Proposition
- 5. Proof of Proposition
- 6. Proof of Proposition
- 7. Proof of Proposition -$\textrm {i})$
- 8. Proof of Proposition -ii)
- 9. Proof of Proposition -iii)
- A. Basics on pseudo-holomorphic curves in symplectizations
- B. Linking properties
- C. Uniqueness and intersections of pseudo-holomorphic curves
Abstract
In this article we study Hamiltonian flows associated to smooth functions $H:\mathbb {R}^4 \to \mathbb {R}$ restricted to energy levels close to critical levels. We assume the existence of a saddle-center equilibrium point $p_c$ in the zero energy level $H^{-1}(0)$. The Hamiltonian function near $p_c$ is assumed to satisfy Moser’s normal form and $p_c$ is assumed to lie in a strictly convex singular subset $S_0$ of $H^{-1}(0)$. Then for all $E>0$ small, the energy level $H^{-1}(E)$ contains a subset $S_E$ near $S_0$, diffeomorphic to the closed $3$-ball, which admits a system of transversal sections $\mathcal {F}_E$, called a $2-3$ foliation. $\mathcal {F}_E$ is a singular foliation of $S_E$ and contains two periodic orbits $P_{2,E}\subset \partial S_E$ and $P_{3,E}\subset S_E\setminus \partial S_E$ as binding orbits. $P_{2,E}$ is the Lyapunoff orbit lying in the center manifold of $p_c$, has Conley-Zehnder index $2$ and spans two rigid planes in $\partial S_E$. $P_{3,E}$ has Conley-Zehnder index $3$ and spans a one parameter family of planes in $S_E \setminus \partial S_E$. A rigid cylinder connecting $P_{3,E}$ to $P_{2,E}$ completes $\mathcal {F}_E$. All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to $P_{2,E}$ in $S_E\setminus \partial S_E$ follows from this foliation.- Salvador Addas-Zanata and Clodoaldo Grotta-Ragazzo, Conservative dynamics: unstable sets for saddle-center loops, J. Differential Equations 197 (2004), no. 1, 118–146. MR 2030151, DOI 10.1016/j.jde.2003.07.010
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