Limit Cycles in a Class of Perturbed Cubic Integrable Non-Hamiltonian Systems with Four Zones

In this paper, we mainly investigate the number of limit cycles of a class of perturbed cubic integrable non-Hamiltonian systems with four zones separated by two nonlinear switching curves. Using the first-order averaging method and complex theory, we establish the lower bound for the maximum number of limit cycles bifurcating from any compact region of the period annulus of the unperturbed system, and also provide the corresponding upper bound for the case where the absolute values of the two key parameters in the model are equal.