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In this paper we research global dynamics and bifurcations of planarpiecewise smooth quadratic quasi--homogeneous but non-homogeneous polynomialdifferential systems. We present sufficient and necessary conditions for theexistence of a center in piecewise smooth quadratic quasi--homogeneous systems.Moreover, the center is global and non-isochronous if it exists, which cannotappear in smooth quadratic quasi-homogeneous systems. Then the globalstructures of piecewise smooth quadratic quasi--homogeneous but non-homogeneoussystems are studied. Finally we investigate limit cycle bifurcations of thepiecewise smooth quadratic quasi-homogeneous center and give the maximal numberof limit cycles bifurcating from the periodic orbits of the center by applyingthe Melnikov method for piecewise smooth near-Hamiltonian systems.

Global dynamics and unfolding of planar piecewise smooth quadratic quasi-homogeneous differential systems