Limit cycles in a class of planar discontinuous piecewise quadratic differential systems with a non-regular line of discontinuity (I)

Abstract

In this paper we study the limit cycles which bifurcate from the periodic orbits of the quadratic uniform isochronous center $\dot{x}=-y+xy$, $\dot{y}=x+y^2$, when this center is perturbed inside the class of all discontinuous piecewise quadratic polynomial differential systems in the plane with two pieces separated by a non-regular line of discontinuity, which is formed by two rays starting from the origin and forming an angle $\alpha =\pi /2$. Using the Chebyshev theory we prove that the maximum number of hyperbolic limit cycles which can bifurcate from these periodic orbits is exactly 8 using the averaging theory of first order. For this class of discontinuous piecewise differential systems we obtain three more limit cycles than the line of discontinuity is regular, i.e., the case of where the two rays form an angle $\alpha =\pi$.