First Integrals vs Limit Cycles

TL;DR

This paper introduces a method using first integrals to identify and bound the number of limit cycles in Liénard systems, providing new insights into their periodic behavior.

Contribution

It presents a novel approach leveraging first integrals and singular ODE solutions to locate and bound limit cycles in Liénard systems.

Findings

Identifies intervals of maximum amplitude limit cycles

Provides an upper bound for the number of limit cycles

Demonstrates the method with examples

Abstract

This paper applies a recent result determining periodic orbits on the basis of first integrals, for Li\'enard systems. By solving a first order ODE with singularities, a crucial result is proved to locate intervals of single and isolated maximum amplitudes periodic orbits (limit cycles). With this result an upper bound for the number of limit cycles is provided. Some examples are presented along with conclusions and future work