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

Continuing the investigation for the piecewise polynomial perturbations of the linear center (x)over dot = -y, (y)over dot = 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 alpha is an element of (0, pi] and 2 pi - alpha. Moreover, unlike the aforementioned work, we allow that the polynomial differential systems in the two sectors have different degrees. Depending on alpha 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 Lienard 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 alpha not equal pi) of the piecewise polynomial perturbations usually leads to more limit cycles than the regular case (i.e. when alpha = pi) where the switching boundary is a straight line. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This study investigates the effects of non-regular switching boundary conditions on piecewise polynomial perturbations of the linear center, finding that non-regular conditions typically lead to more limit cycles compared to regular conditions. Through the averaging method, an upper bound for the maximum number of limit cycles is provided, with some characteristics of systems with different degrees presented.