Limit cycles in piecewise polynomial systems allowing a non-regular switching boundary

Abstract

Continuing the investigation for the piecewise polynomial perturbations of the linear center ẋ=−y,ẏ=x from Buzzi et al. (2018) for the case where the switching boundary is a straight line, in this paper we allow that the switching boundary is non-regular, i.e. we consider a switching boundary which separates the plane into two angular sectors with angles α∈(0,π] and 2π−α. Moreover, unlike the aforementioned work, we allow that the polynomial differential systems in the two sectors have different degrees. Depending on α and for arbitrary given degrees we provide an upper bound for the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center using the averaging method up to any order. This upper bound is reached for the first two orders. On the other hand, we pay attention to the perturbation of the linear center inside this class of piecewise polynomial Liénard systems and give some better upper bounds in comparison with the one obtained in the general piecewise polynomial perturbations. Again our results imply that the non-regular switching boundary (i.e. when α≠π) of the piecewise polynomial perturbations usually leads to more limit cycles than the regular case (i.e. when α=π) where the switching boundary is a straight line.