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

Abstract

It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in $\mathbb{R}^2$ is $. In contrast here we consider discontinuous differential systems in $\mathbb{R}^2$ defined in two half-planes separated by a straight line. In one half plane we have a general linear center at the origin of $\mathbb{R}^2$, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of $\mathbb{R}^2$. 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.