Limit Cycles in a Class of Planar Discontinuous Piecewise Quadratic Differential Systems with a Non-regular Line of Discontinuity (II)

Abstract

In our previous work, we have studied the limit cycles for a class of discontinuous piecewise quadratic polynomial differential systems with a non-regular line of discontinuity, which is formed by two rays starting from the origin and forming an angle α=π/2. The unperturbed system is the quadratic uniform isochronous center x˙=-y+xy, y˙=x+y2 with a family of periodic orbits surrounding the origin. In this paper, we continue to investigate this kind of piecewise differential systems, but now the angle between the two rays is α∈(0,π/2)∪[3π/2,2π). Using the Chebyshev theory, we prove that the maximum number of hyperbolic limit cycles that can bifurcate from these periodic orbits using the averaging theory of first order is exactly 8 for α∈(0,π/2)∪[3π/2,2π). Together with our previous work, which concerns on the case of α=π/2, we can conclude that using the averaging theory of first order the maximum number of hyperbolic limit cycles is exactly 8, when this quadratic center is perturbed inside the above-mentioned classes separated by a non-regular line of discontinuity with α∈(0,π/2]∪[3π/2,2π).