Bifurcation delay - the case of the sequence: stable focus - unstable focus - unstable node

TL;DR

This paper investigates the continuation of bifurcation delay in a two-dimensional vector field family, focusing on the transition from a focus to a node after a Hopf bifurcation.

Contribution

It analyzes the behavior of bifurcation delay beyond the focus-node bifurcation in a specific vector field family with sequential stationary point types.

Findings

Bifurcation delay persists after the focus-node transition.

The delay behavior depends on the slow parameter variation.

The study extends understanding of delay phenomena in dynamical systems.

Abstract

Let us give a two dimensional family of real vector fields. We suppose that there exists a stationary point where the linearized vector field has successively a stable focus, an unstable focus and an unstable node. When the parameter moves slowly, a bifurcation delay appears due to the Hopf bifurcation. The studied question in this article is the continuation of the delay after the focus-node bifurcation.