Limit Cycles of a Class of Discontinuous Piecewise Differential Systems Separated by the Curve y = xn Via Averaging Theory

Recently there is increasing interest in studying the limit cycles of the piecewise differential systems due to their many applications. In this paper we prove that the linear system [Formula: see text], [Formula: see text], can produce at most seven crossing limit cycles for [Formula: see text] using the averaging theory of first order, where the bounds [Formula: see text] for [Formula: see text] even and the bounds [Formula: see text] for [Formula: see text] odd are reachable, when it is perturbed by discontinuous piecewise polynomials formed by two pieces separated by the curve [Formula: see text] ([Formula: see text]), and having in each piece a quadratic polynomial differential system. Using the averaging theory of second order the perturbed system can be chosen in such way that it has [Formula: see text] or [Formula: see text] crossing limit cycles for [Formula: see text] even and, furthermore, under a particular condition we prove that the number of crossing limit cycles does not exceed [Formula: see text] (resp., [Formula: see text]) for [Formula: see text] even (resp., [Formula: see text] even). The averaging theory of second order produces the same number of crossing limit cycles as the averaging theory of first order if [Formula: see text] is odd. The main tools for proving our results are the new averaging theory developed for studying the crossing limit cycles of the discontinuous piecewise differential systems, and the theory for studying the zeros of a function using the extended Chebyshev systems.