Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential systems

Abstract

In this paper, we consider the quadratic isochronous centers perturbed inside piecewise polynomial differential systems of arbitrary degree n with the straight line of discontinuity x=0. The main concerns are the number of zeros of the first order Melnikov functions and the estimate of the number of limit cycles bifurcating from the period annuli. For quadratic isochronous centers S1, S2 and S3, we will provide a sharp upper bound for the number of zeros of the first order Melnikov functions. For quadratic isochronous center S4, we give a rough estimate. However, when the problem is reduced to perturbations inside polynomial differential systems, our result for S4 will improve that in Li et al. (2000) [12] significantly. Moreover, we will reveal out some equivalence between the first order Melnikov function method and the first order averaging method for investigating the number of limit cycles of piecewise polynomial differential systems.