Bifurcation of limit cycles by perturbing piecewise smooth integrable differential systems with four zones

This paper deals with the problem of limit cycle bifurcations for piecewise smooth integrable differential systems with four zones. When the unperturbed system has a family of periodic orbits, the first order Melnikov function is derived which can be used to study the number of limit cycles bifurcated from the periodic orbits. As an application, using the first order Melnikov function and Picard-Fuchs equation, we obtain an upper bound of the number of bifurcated limit cycles of a concrete piecewise smooth differential system.

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