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Systems of transversal sections near critical energy levels of Hamiltonian systems in $\mathbb{R}^4$

Naiara V. de Paulo, Pedro A. S. Salomão

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.

Systems of transversal sections near critical energy levels of Hamiltonian systems in $\mathbb{R}^4$

Abstract

In this article we study Hamiltonian flows associated to smooth functions restricted to energy levels close to critical levels. We assume the existence of a saddle-center equilibrium point in the zero energy level . The Hamiltonian function near is assumed to satisfy Moser's normal form and is assumed to lie in a strictly convex singular subset of . Then for all small, the energy level contains a subset near , diffeomorphic to the closed -ball, which admits a system of transversal sections , called a foliation. is a singular foliation of and contains two periodic orbits and as binding orbits. is the Lyapunoff orbit lying in the center manifold of , has Conley-Zehnder index and spans two rigid planes in . has Conley-Zehnder index and spans a one parameter family of planes in . A rigid cylinder connecting to completes . All regular leaves are transverse to the Hamiltonian vector field. The existence of a homoclinic orbit to in follows from this foliation.

Paper Structure

This paper contains 18 sections, 52 theorems, 345 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Saddle-center equilibrium points
  3. Strictly convex subsets of the critical energy level
  4. $2-3$ foliations
  5. Main statement
  6. Applications
  7. An open question
  8. Proof of the main statement
  9. Proof of Proposition \ref{['prop_step1']}
  10. Proof of Proposition \ref{['prop_step2']}
  11. Proof of Proposition \ref{['prop_step3']}
  12. Proof of Proposition \ref{['prop_step4']}
  13. Proof of Proposition \ref{['prop_step5']}-${\rm i})$
  14. Proof of Proposition \ref{['prop_step5']}- ii)
  15. Proof of Proposition \ref{['prop_step5']}- iii)
  16. ...and 3 more sections

Key Result

Lemma 1.4

For all $(E,\delta)\neq (0,0),$ with $E,\delta \geq 0$ sufficiently small, the set is an embedded $2$-sphere in $K^{-1}(E).$

Figures (14)

  • Figure 1.1: Local behavior of the flow near a saddle-center equilibrium point projected into the planes $(q_1,p_1)$ and $(q_2,p_2)$.
  • Figure 1.2: The projections of $K^{-1}(E)$ into the plane $(q_1,p_1)$ for $E<0$, $E=0$ and $E>0$.
  • Figure 1.3: The sphere-like singular subset $S_0=\varphi(B_0^\delta) \cup B_\delta \subset H^{-1}(0).$
  • Figure 1.4: The embedded closed $3$-ball $S_E \subset H^{-1}(E),$$E>0$ small.
  • Figure 1.5: The embedded $2$-sphere $\partial S_E =\varphi(N_E^0)\subset H^{-1}(E),$$E>0$, and its hemispheres $U_{1,E}$ and $U_{2,E}$. The arrows point in the direction of the Hamiltonian vector field.
  • ...and 9 more figures

Theorems & Definitions (98)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9
  • ...and 88 more