On the torus bifurcation in averaging theory

TL;DR

This paper uses averaging theory to identify conditions for torus bifurcations in 2D nonautonomous systems and applies these findings to 3D vector fields, advancing understanding of complex bifurcation phenomena.

Contribution

It introduces generic conditions on averaged functions that guarantee the occurrence of Neimark-Sacker bifurcations in parameter spaces, linking them to torus bifurcations.

Findings

Derived conditions for Neimark-Sacker bifurcation in averaged systems

Established a connection between bifurcations in 2D and 3D vector fields

Provided a framework for analyzing torus bifurcations via averaging theory

Abstract

In this paper, we take advantage of the averaging theory to investigate a torus bifurcation in two-parameter families of 2D nonautonomous differential equations. Our strategy consists in looking for generic conditions on the averaged functions that ensure the existence of a curve in the parameter space characterized by a Neimark-Sacker bifurcation in the corresponding Poincar\'{e} map. A Neimark-Sacker bifurcation for planar maps consists in the birth of an invariant closed curve from a fixed point, as the fixed point changes stability. In addition, we apply our results to study a torus bifurcation in a family of 3D vector fields.