Piecewise Smooth Perturbation For Two Classes Of Differential Systems

As we all know, in the qualitative theory of differential systems, investigating the stability, existence, the number and the distribution of limit cycles has the important practical significance and theoretical value. A certain degree planar polynomial differ-ential system, one can use bifurcation theory to study the number of limit cycles. A large number of studies have been made on the smooth perturbation problem of periodic solutions of polynomial systems. In this paper, we mainly discuss the piecewise smooth perturbation problem for the periodic solutions of a smooth polynomial system.The first chapter is the preface, in which we mainly introduce the background and known results of the limit cycles of polynomial differential systems and briefly illustrate the main research contents of the thesis.In chapter 2, we study the number of limit cycles that bifurcate from the periodic so-lutions of a quadratic differential system with an isochronous center, when it is perturbed by piecewise smooth quadratic polynomials. By using first order Melnikov functions to this system, it is proved that 6 limit cycles can bifurcate from the period annulus. This result improves Conclusion of the literature [23]. Compared with the result in [23], two more limit cycle is foundIn chapter 3, we study the number of limit cycles that bifurcate from the periodic solutions of a cubic differential system, when it is perturbed by piecewise smooth cubic polynomials. By using first order Melnikov functions to this system, it is proved that 8 limit cycles can bifurcate from the period annulus. The result shows that piecewise cubic polynomials perturbation cubic differential system can have 4 more limits cycles than corresponding cubic polynomials perturbation...