Global stability analysis of birhythmicity in a self-sustained oscillator

TL;DR

This paper investigates the global stability of birhythmicity in a modified van der Pol oscillator under random noise, revealing how nonlinear coefficients influence attractor stability and escape times.

Contribution

It introduces a method to measure global stability of birhythmicity using mean escape times and analyzes the effects of nonlinear parameters and noise on system stability.

Findings

Different trapping barriers for the two frequencies affect system stability.

System nearly symmetric within a narrow parameter range.

Escape time varies linearly with inverse noise intensity.

Abstract

We analyze global stability properties of birhythmicity in a self-sustained system with random excitations. The model is a multi-limit cycles variation of the van der Pol oscillatorintroduced to analyze enzymatic substrate reactions in brain waves. We show that the two frequencies are strongly influenced by the nonlinear coefficients $\alpha$ and $\beta$. With a random excitation, such as a Gaussian white noise, the attractor's global stability is measured by the mean escape time $\tau$ from one limit-cycle. An effective activation energy barrier is obtained by the slope of the linear part of the variation of the escape time $\tau$ versus the inverse noise-intensity 1/D. We find that the trapping barriers of the two frequencies can be very different, thus leaving the system on the same attractor for an overwhelming time. However, we also find that the system is nearly symmetric in a narrow range of the parameters.