Lower bounds for the number of limit cycles in a generalized Rayleigh-Li\'enard oscillator

TL;DR

This paper establishes lower bounds on the number of limit cycles in a generalized Rayleigh-Liénard oscillator using Lyapunov constants and Melnikov method, demonstrating the existence of up to twelve limit cycles under certain conditions.

Contribution

It provides new lower bounds for limit cycles in a generalized oscillator, explicitly exhibiting the method to find these cycles using Lyapunov constants and Melnikov method.

Findings

Up to twelve limit cycles can exist under certain parameter conditions.

The approach involves parameter sign changes and Poincaré-Bendixson Theorem.

Explicit method for obtaining limit cycles is demonstrated.

Abstract

In this paper a generalized Rayleigh-Li\'enard oscillator is consider and lower bounds for the number of limit cycles bifurcating from weak focus equilibria and saddle connections are provided. By assuming some open conditions on the parameters of the considered system the existence of up to twelve limit cycles is provided. More precisely, the approach consists in perform suitable changes in the sign of some specific parameters and apply Poincar\'e-Bendixson Theorem for assure the existence of limit cycles. In particular, the method for obtaining the limit cycles through the referred approach is explicitly exhibited. The main techniques applied in this study are the Lyapunov constants and the Melnikov method.