Systematic designing of bi-rhythmic and tri-rhythmic models in families of Van der Pol and Rayleigh oscillators

TL;DR

This paper presents a systematic method for designing Van der Pol and Rayleigh oscillator families with specific multiple limit cycles, using a general LLS system framework and numerical simulations.

Contribution

It introduces a scheme for systematically designing oscillators with desired multiple limit cycles within the LLS system class, expanding control over nonlinear dynamics.

Findings

Successfully designed bi-rhythmic and tri-rhythmic oscillator models.

Demonstrated the parameter space search for multiple limit cycles.

Extended the approach to higher order variants.

Abstract

Van der Pol and Rayleigh oscillators are two traditional paradigms of nonlinear dynamics. They can be subsumed into a general form of Li\'enard--Levinson--Smith(LLS) system. Based on a recipe for finding out maximum number of limit cycles possible for a class of LLS oscillator, we propose here a scheme for systematic designing of generalised Rayleigh and Van der Pol families of oscillators with a desired number of multiple limit cycles. Numerical simulations are explicitly carried out for systematic search of the parameter space for bi-rhythmic and tri-rhythmic systems and their higher order variants.