Poincare index and periodic solutions of perturbed autonomous systems

TL;DR

This paper introduces a novel approach using the Poincare index to analyze bifurcations of periodic solutions in perturbed autonomous systems, relaxing the nondegeneracy condition and offering new geometric insights.

Contribution

It applies the Poincare index to study periodic solutions, removing the need for nondegeneracy and enhancing geometric understanding of bifurcations in autonomous systems.

Findings

Poincare index provides a new method for analyzing bifurcations.

The approach relaxes the nondegeneracy requirement.

It offers additional geometric properties of solutions.

Abstract

The basic tool of classical results by Malkin and Melnikov on bifurcation of periodic solutions from nondegenerate cycles of autonomous systems with periodic perturbations is an implicit function theorem. In this paper the Poincare index is used to avoid the requirement of nondegeneracity for the unperturbed cycles and to provide additional geometrical properties of periodic solutions of the perturbed system.