Stability of Periodic Orbits by Conley-Zehnder index theory

TL;DR

This paper establishes a criterion for the strong stability of low-dimensional Hamiltonian systems using the Conley-Zehnder index, with applications to Mathieu equations and harmonic oscillations.

Contribution

It provides a necessary and sufficient condition for stability based on the iterates of closed orbits and the Conley-Zehnder index, advancing stability analysis methods.

Findings

Derived a stability criterion using Conley-Zehnder index

Applied the criterion to Mathieu equations

Analyzed stable harmonic oscillations in pendulum systems

Abstract

We give a necessary and sufficient condition for strong stability of low dimensional Hamiltonian systems, in terms of the iterates of a closed orbit and the Conley-Zehnder index. Applications to Mathieu equation and stable harmonic oscillations for forced pendulum type equations are considered as applications of the main result.