The Number of Limit Cycles Bifurcating from a Degenerate Center of Piecewise Smooth Differential Systems

Abstract

For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate [Formula: see text]-center, derived from a nonsmooth harmonic oscillator model, to degenerate [Formula: see text]-center. Our study involves some new computational treatments. The main tools are the generalized polar coordinate change and the generalized Lyapunov polar coordinate change together with an averaging theory for one-dimensional piecewise smooth differential equations. Finally, we present two Maple programs for computing the averaging functions and consequently the biggest lower bound on the maximum number of limit cycles of degenerate [Formula: see text]-center under general polynomial perturbations of degree [Formula: see text].