A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system

In this paper we study a class of planar piecewise smooth quadratic integrable non-Hamiltonian systems, which have a center. By using the averaging method, we give an estimation of the number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n. Our estimation is linear depending on n and it is at least twice the associated estimation of smooth systems.