On the Number of Limit Cycles of Planar Piecewise Smooth Quadratic Systems with Focus-Parabolic Type Critical Point

In this paper we investigate the number of small amplitude limit cycles bifurcated from a class of planar piecewise smooth quadratic systems with focus-parabolic type critical point having exactly one switching line. By considering the Lyapunov constants of the system, we obtain some conditions under which (0, 0) is a center or a focus of order eleven. We prove that at least ten limit cycles can bifurcate from (0, 0), which is a new lower bound of the cyclicity of piecewise smooth quadratic systems with focus-parabolic type critical point having exactly one switching line. For four of the center conditions, we prove that at least six limit cycles can bifurcate from (0, 0) without destroying the singularity by higher order perturbations techniques, one more than that obtained by considering the linear parts of the perturbed Lyapunov constants. Finally, the center-focus problem for a subclass of the system is completely solved.