Limit Cycles and Stability Analysis in 3-Piecewise Linear Differential Systems with a Double Heteroclinic Loop

This paper is devoted to studying lower and upper bounds of the number of limit cycles in a class of 3-piecewise linear differential systems with a double heteroclinic loop under perturbations of polynomials with degree n. By using the first-order Melnikov function, we obtain that [Formula: see text] limit cycles can appear near two centers and the double heteroclinic loop, respectively, and that there are at most [Formula: see text] limit cycles bifurcating from two period annuluses between two centers and the double heteroclinic loop up to the first order in [Formula: see text]. In particular, for [Formula: see text], we consider a 3-piecewise linear [Formula: see text]-equivariant system and, by using the stability-changing method, obtain a total of 10 limit cycles, four of which are alien limit cycles with two large ones surrounding two symmetric smaller ones.