Limit cycles on rigid piecewise smooth dynamical systems governed by even polynomials

Abstract

In this work, we establish an upper bound for the number of crossing limit cycles in a class of piecewise smooth dynamical systems. The system is formed by a linear rigid center and a rigid center governed by a homogeneous polynomial of even degree n, separated by the straight line $x=0$. Our results complement the work of [1], which addressed the odd-degree case. Specifically, we prove that if the parameters satisfy $d_2=M_2$, the system admits at most $(n-2)/2$ limit cycles. Furthermore, for the specific case $n=4$, assuming d₂ ≠ M₂ and $d_2=0$, we show that the system has at most one limit cycle, and this upper bound is attained. This study advances the analysis of this family of systems by covering the even-degree case under certain conditions on the affine transformation.