Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems

TL;DR

This paper extends the theory of Hopf bifurcations to higher-dimensional discrete dynamical systems, providing sharper theorems and topological insights into attractors and bifurcations.

Contribution

It introduces generalized Hopf bifurcation theorems for discrete systems using topological methods, broadening understanding of attractors in higher dimensions.

Findings

Sharper Hopf bifurcation theorems for fixed points and attractors

Topological techniques based on concentricity of manifolds

General results for attractors in n-dimensional manifolds

Abstract

In this paper we study generalized Poincar\'e-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in n-dimensional manifolds satisfying some suitable conditions. This result allows us to obtain sharper Hopf bifurcation theorems for fixed points in the general case and other attractors in low dimensional manifolds. Topological techniques based on the notion of concentricity of manifolds play a substantial role in the paper.