Bifurcation of Limit Cycles from a Class of Piecewise Smooth Systems with Two Vertical Straight Lines of Singularity

Perturbing the following systems (x,y) = (-y(1 + by),x(1 + by))x > 0,(-y(1 + ax), x(1 + ax)) x < 0, having a linear center with two vertical straight lines (ab0) of singularity inside the class of all piecewise polynomials of degree n, we give the estimate of the maximum number H(n) of limit cycles bifurcating from the period annulus around the center based on the argument principle and the first order averaging theory. It shows that H(n) is at least [n/2] + 1 bigger than the maximum number of limit cycles bifurcating from the period annulus around the linear center with two parallel straight lines of singularity in [Li & Liu, 2015].