Limit cycles bifurcate from centers of discontinuous quadratic systems

Like for smooth quadratic systems, it is important to determine the maximum order of a fine focus and the cyclicity of discontinuous quadratic systems. Previously, examples of discontinuous quadratic systems with five limit cycles bifurcated from a fine focus of order 5 have been constructed. In this paper we construct a class of discontinuous quadratic systems with a fine focus of order 9. In addition, by using a method similar to that developed by C. Christopher for smooth systems, which allows one to estimate the cyclicity just from the lower order terms of Lyapunov constants, we show that the cyclicity of discontinuous quadratic systems is at least 9, thus improving on previous results.