Some results on homoclinic and heteroclinic connections in planar systems

TL;DR

This paper develops an algebraic method to bound bifurcation sets in planar systems with limit cycles, demonstrated on quadratic families including the Bogdanov-Takens system, providing explicit bifurcation curves without Melnikov functions.

Contribution

It introduces a novel algebraic approach to estimate bifurcation sets in planar systems, avoiding Melnikov function evaluations, and applies it to quadratic systems including the Bogdanov-Takens system.

Findings

Derived explicit algebraic bounds for bifurcation sets.

Obtained the bifurcation curve for small n in the Bogdanov-Takens system.

Produced new algebraic terms for bifurcation curves without Melnikov functions.

Abstract

Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $\Phi(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${\Phi(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/45294865}n^{3/2}- {2352961656/11108339166925} n^2+O(n^{5/2})$. We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions.