Inverse Jacobian multipliers and Hopf bifurcation on center manifolds

TL;DR

This paper investigates higher-dimensional differential systems with a two-dimensional center manifold, characterizing inverse Jacobian multipliers and analyzing Hopf bifurcation to understand system cyclicity.

Contribution

It provides a characterization of analytic and smooth inverse Jacobian multipliers in systems with a center or focus, linking their properties to Hopf bifurcation analysis.

Findings

Existence criteria for inverse Jacobian multipliers around the origin.

Relationship between inverse Jacobian multipliers and system cyclicity.

Application of vanishing multiplicity to Hopf bifurcation analysis.

Abstract

In this paper we consider a class of higher dimensional differential systems in $\mathbb R^n$ which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of either analytic or $C^\infty$ inverse Jacobian multipliers of the systems around the origin, which is either a center or a focus on the center manifold. Later we study the cyclicity of the system at the origin through Hopf bifurcation by using the vanishing multiplicity of the inverse Jacobian multiplier.