Transversal conics and the existence of limit cycles

TL;DR

This paper introduces a method using transversal conics to locate and analyze limit cycles in planar polynomial differential systems, providing bounds and regions for their existence and bifurcations.

Contribution

It presents a novel approach to construct Poincaré--Bendixson regions with transversal conics for better understanding limit cycles.

Findings

Method effectively locates limit cycles in various systems.

Provides bounds for bifurcation parameters.

Demonstrates the approach on known systems with different bifurcation features.

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 conics. We present several examples of known systems in the literature showing different features about limit cycles: hyperbolicity, Hopf bifurcation, sky-blue bifurcation, rotated vector fields, \ldots for which the obtained Poincar\'e--Bendixson region allows to locate the limit cycles. Our method gives bounds for the bifurcation values of parametrical families of planar vector fields and intervals of existence of limit cycles.