Limit Cycles in Discontinuous Planar Piecewise Differential Systems with Multiple Nonlinear Switching Curves

In this paper, we investigate the maximum number of limit cycles that can bifurcate from the periodic annulus of the linear center for discontinuous piecewise quadratic polynomial differential systems with four zones separated by two nonlinear curves \(y=\pm x^{2}\) $y=\pm x^2$ . By analyzing the first order averaged function, we prove that at most 7 crossing limit cycles can produce from periodic annulus of the linear center, and the upper bound is reached. In addition, under particular conditions, we obtain that at least 8 crossing limit cycles can bifurcate from periodic annulus in this class of systems by using the second order averaged function. Our results show that multiple switching curves increase the number of crossing limit cycles in comparison with the case where there is only a single switching curve.