Recherche et \'etude de la stabilit\'e du cycle limite pour l'oscillateur de Rayleigh

TL;DR

This paper investigates the Rayleigh autonomous oscillator's limit cycle and its stability, using analytical methods and numerical simulations to reveal bifurcation phenomena.

Contribution

The study introduces a novel analytical approach to identify the limit cycle of the Rayleigh oscillator and analyzes its stability using the method of multiple scales.

Findings

Identification of the limit cycle using Poincaré-Bendixson theorem

Analytical and numerical confirmation of bifurcation of PAH

Demonstration of the oscillator's single fixed point

Abstract

In this paper, we studied Rayleigh autonomous oscillator by searching its limit cycle and by studying the stability of the cycle. Through this study, we studied the fixed points of the equation of Rayleigh oscillator and we realized that in reality this oscillator has only one fixed point. What appears new according to our point of view is that we found the cycle limit of the oscillator using a form of Poincar\'e-Bendixson theorem. Then, we studied the stability of this limit cycle by the method of multiple scales. Finally, the most important is that in this paper we have shown analytically and confirmed by numerical simulation using Mathematica that the Rayleigh oscillator exhibits a bifurcation of PAH