Generalization of Lyapunov Center Theorem for Hamiltonian systems via normal forms theory

TL;DR

This paper extends the Lyapunov Center Theorem to a broader class of Hamiltonian systems by employing normal forms theory and bifurcation analysis to establish conditions for periodic solutions near equilibria.

Contribution

It generalizes existing theorems by formulating new sufficient conditions for nonstationary periodic solutions in autonomous Hamiltonian systems using normal forms and bifurcation theory.

Findings

Established new sufficient conditions for periodic solutions near equilibria.

Unified previous theorems under broader assumptions.

Applied normal forms theory to analyze Hamiltonian matrices.

Abstract

In this article we formulate and prove sufficient conditions for the existence of trajectories of nonstationary periodic solutions of autonomous Hamiltonian systems in a neighbourhood of equilibria. It is worth pointing out that assumptions of some well-known theorems imply that of our main results. We obtain our results with the use of the theory of normal forms for Hamiltonian matrices and global bifurcation theory for autonomous Hamiltonian systems.