Hopf bifurcations in fast/slow systems with rate-dependent tipping

TL;DR

This paper investigates how rate-dependent tipping occurs in fast/slow systems, revealing a Hopf bifurcation that leads to the growth of a limit cycle, with applications to the van der Pol oscillator.

Contribution

It demonstrates the occurrence of Hopf bifurcations in rate-dependent tipping within fast/slow systems, providing a new perspective on the transition dynamics.

Findings

Hopf bifurcation occurs as the rate parameter increases.

Growth of a limit cycle signals the transition from tracking to tipping.

Application to a forced van der Pol oscillator illustrates the phenomenon.

Abstract

We analyze rate-dependent tipping in a fast/slow system with an equilibrium near the fold of a critical manifiold. We find a Hopf bifurcation as the rate parameter increases in the reduced co-moving system. This implies the growth of a limit cycle as the system changes from tracking a quasi-static equilibrium to tipping. Rather than trajectories diverging at a critical rate, they continue to track the quasi-static equilibrium in a spiral corresponding to an emerging limit cycle at the Hopf bifurcation. We apply the same analysis to a forced van der Pol oscillator to show this phenomenon in a familiar system where the growth of this limit cycle is well understood.