A class of reversible quadratic systems with piecewise polynomial perturbations

Abstract

This paper investigates a class of reversible quadratic systems perturbed inside piecewise polynomial differential systems of arbitrary degree n. All possible phase portraits of the reversible quadratic systems on the plane are obtained by analytic techniques. Then, we present the algebraic structure of its corresponding first-order Melnikov function, which can be used to study the limit cycle bifurcation problem. In this direction, we use it to investigate the Hopf bifurcation of the perturbed systems, and obtain an upper bound for the Hopf cyclicity. Meanwhile, we employ it to discuss the maximal number of limit cycles emerging from the periodic orbits around the origin for a special case and also provide its upper bound.