Piecewise Di erential System

Abstract

In this thesis, we rst gave essential information such as de nitions, lemmas, and theorems used in our study. Second, we have presented the averaging theory for computing limit cycles for the systems, focus-focus, focus-center, with the straight separation line y = 0; we proved that there are at most 1, 2, or 3 limit cycles, and the systems, center-center can not have limit cycles. Next, we studied the limit cycles of planar piecewise di erential systems formed by quadratic systems and linear centers. We proved that these piecewise systems, with the straight separation line y = 0; have a continuum of periodic orbits and can have at most 2, 3, 4, 5, and 8 limit cycles. Finally, we have tackled the number of periodic solutions of the piecewise di erential systems formed by the linear center and isochronous cubic system separated by the straight-line x = 0 and y = ax + b by treating the two cases as continuous and discontinuous. We proved that piecewise systems have three solutions with a long formula. 39