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A Poincaré-Birkhoff theorem for multivalued successor maps with applications to periodic superlinear Hamiltonian systems

Guglielmo Feltrin, Alessandro Fonda, Andrea Sfecci

Abstract

We provide a new version of the Poincaré-Birkhoff theorem for possibly multivalued successor maps associated with planar non-autonomous Hamiltonian systems. As an application, we prove the existence of periodic and subharmonic solutions of the scalar second order equation $\ddot x + λg(t,x) = 0$, for $λ>0$ sufficiently small, with $g(t,x)$ having a superlinear growth at infinity, without requiring the existence of an equilibrium point.

A Poincaré-Birkhoff theorem for multivalued successor maps with applications to periodic superlinear Hamiltonian systems

Abstract

We provide a new version of the Poincaré-Birkhoff theorem for possibly multivalued successor maps associated with planar non-autonomous Hamiltonian systems. As an application, we prove the existence of periodic and subharmonic solutions of the scalar second order equation , for sufficiently small, with having a superlinear growth at infinity, without requiring the existence of an equilibrium point.

Paper Structure

This paper contains 11 sections, 15 theorems, 109 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The successor map for Hamiltonian systems
  3. From times to angles
  4. Successor map vs. Poincaré map
  5. A Poincaré--Birkhoff theorem for multivalued successor maps
  6. Sufficient conditions for the existence of the successor map
  7. A superlinear differential equation
  8. Step 1. The construction of some guiding curves.
  9. Step 1a. Entering solution guiding curve.
  10. Step 1b. Exiting solution guiding curve.
  11. Step 2. The modified problem.

Key Result

Corollary 1.1

For any integer $N$, there exists $\lambda_N>0$ such that, for every $\lambda \in\mathopen{]}0,\lambda_N\mathclose{]}$, equation ela has at least $N$ periodic solutions.

Figures (4)

  • Figure 1: Qualitative representation of hypotheses \ref{['hp-A-4']} (concerning the area colored in light gray) and \ref{['hp-A-5']} (concerning the area colored in dark gray).
  • Figure 2: Qualitative representation of the statement of Theorem \ref{['successor_thm']}.
  • Figure 3: a) The set $\bighat {\mathcal{C}}_j$ delimiting the interior regions $\bighat{\mathcal{I}}_j$ and the exterior regions $\bighat{\mathcal{E}}_j$. The arrows suggest the direction of the vector field associated with \ref{['HS-lambda']}, see Lemma \ref{['lem-xx']}. b) A sketch of Proposition \ref{['prop-xx']} in the case $m=1$.
  • Figure 4: a) The set $\bigcheck {\mathcal{C}}_j$ delimiting the interior regions $\bigcheck{\mathcal{I}}_j$ and the exterior regions $\bigcheck{\mathcal{E}}_j$. The arrows suggest the direction of the vector field associated with \ref{['HS-lambda']}, see Lemma \ref{['lem-xx2']}. b) A sketch of Proposition \ref{['prop-xx2']} in the case $m=1$.

Theorems & Definitions (33)

  • Corollary 1.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Remark 2.5
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 23 more