Limit cycles appearing from the perturbation of differential systems with multiple switching curves

Abstract This paper deals with the problem of limit cycle bifurcations for a piecewise near-Hamilton system with four regions separated by algebraic curves y = ± x 2 . By analyzing the obtained first order Melnikov function, we give an upper bound of the number of limit cycles which bifurcate from the period annulus around the origin under nth degree polynomial perturbations. In the case n = 1 , we obtain that at least 4 (resp. 3) limit cycles can bifurcate from the period annulus if the switching curves are y = ± x 2 (resp. y = x 2 or y = − x 2 ). The results also show that the number of switching curves affects the number of limit cycles.