Bifurcations of heteroclinic contours in two-parameter planar systems: Overview and explicit examples

TL;DR

This paper reviews bifurcations of heteroclinic contours in planar systems with two saddles, introduces explicit polynomial examples, and demonstrates their bifurcation behavior using computational tools.

Contribution

It provides an overview of heteroclinic contour bifurcations and introduces new explicit polynomial systems illustrating these phenomena.

Findings

Explicit polynomial systems with heteroclinic contours derived

Bifurcation phenomena confirmed using MatCont software

Series of heteroclinic connections observed in examples

Abstract

Smooth planar vector fields containing two hyperbolic saddles may possess contours formed by heteroclinic connections between these saddles. We present an overview of the bifurcations of these contours based on papers by J.W. Reyn and A.V. Dukov. Additionally, two new explicit polynomial systems containing such contours are derived, which are studied using the bifurcation software MatCont and are shown to exhibit the theoretically predicted phenomena, including series of heteroclinic connections.