Weak Centers And Local Bifurcations Of Critical Periods For Liénard Equations

There are two important types of problems for center-type differential equations, one of which is the problem of weak foci and the Hopf bifurcations, while another is the problem of weak centers and bifurcations of critical periods. In this paper we discuss problems of weak centers for nondegenerate differential equations, which are organized as following:In the preface of this paper, we introduce the basic concepts of center-type differential equations, including definitions of equilibria, weak foci, centers, weak centers of order k, isochronous centers and bifurcations of critical periods and the Period Coefficient Lemma. The development of researching on problems of weak foci and problems of weak centers are introduced, and our main work and results in this paper are summarized.In chapter 2, we introduce the basic method to analyze zeros of high degree polynomials with computer algebra system, including pseudo division andresultant.In chapter 3, it is devoted to the centers and isochronous centers of polynomial Lienard equations. We analyze former's centers conditions for polynomial Lienard equations, give direct conditions of a nondegenerate center at the origin with elimination theory, and give an algorithm for general situations with algebraic elimination methods of polynomials. Based on former's criteria of isochronous centers for centers, we analyze parameter conditions of isochronous centers. Based on former's criteria of isochronous centers for centers, we detail...