Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems

It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in ℝ2 is 3. In contrast here we consider discontinuous differential systems in ℝ2 defined in two half-planes separated by a straight line. In one half plane we have a general linear center at the origin of ℝ2, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of ℝ2. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function.