Proving the existence of numerically detected planar limit cycles

TL;DR

This paper introduces a method to rigorously prove the existence and precise location of limit cycles in planar polynomial differential systems, bridging numerical detection with mathematical proof.

Contribution

It presents a novel approach using transversal curves to construct Poincaré--Bendixson regions, enabling the proof of numerically detected limit cycles in various systems.

Findings

Successfully proved the existence of limit cycles in the Brusselator and Lie9nard systems.

Achieved precise localization of limit cycles in phase space.

Provided sharp bounds for bifurcation values in saddle-node bifurcations.

Abstract

This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincar\'e--Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Li\'{e}nard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov system.