Center-Focus Determination And Bifurcation Of Limit Cycles Of The Equator For Planar Polynomial Differential Systems

This thesis is devoted to center-focus determination and bifurcation of limit cycles from the equator in planar polynomial differential systems. It consists of seven chapters.In Chapter 1, the background and present conditions are introduced and summarized for the study of center-focus determination and bifurcation of limit cycles of the equator for planar polynomial differential systems.In Chapter 2, by studying the computation of the quantities of singular point of the original of the following complex autonomous polynomial differential systemtwo linear recursion formulas for the computation of quantities of singular point of system (1) are obtained. Applicable formulas are presented unitedly for the computation of focus quantities and saddle quantities, which play an important role in center-focus determination and bifurcation of limit cycles in real planar polynomial differential systems. So far, there is no effective method for the computation of focus quantities and saddle quantities. Compared with the work of other authors, complex nonlinear integrating operation and solving multivariate linear equations are avoided in computation, which are necessary in more usual approaches. The calculation and simplification of focus quantities can be readily done with these formulas and computer symbol operation systems. A formula algorithm for saddle quantities is given and used for the first time. With forcing only addition, subtraction, multiplication and division to the coefficients of system (1), the m -th quantity of singular point can be deduced and expressed directly in the coefficient of the system. As the constants involved are rational, there is no error in the computation. The first 3 focus quantities and the first 3 saddle quantities are derived simply and quickly with the formulas for real planar quadratic systems. At the same time, the first 8 focus quantities, center conditions and center integral are given for a cubic system and a computational example of cubic systems with 10 saddle quantities presented for the first time.From Chapter 3 to Chapter 6, the center-focus determination and bifurcation of the equator are studied for real planar odd polynomial differential systems. In chapter 3, the stability and bifurcation of the equator of a cubic system are investigated. By translating this real cubic system into a complex planar system, an applicable linear algebraic recursion formula of the equator and the first 6 quantities of the equator are given. The integrability conditions and coefficient conditions for the appearance of 5 and 6 limit cycles from the neighborhood of the equator are obtained. An example of cubic system with 6 limit cycles bifurcating from the equator is given for the first time. Without constructing Poincare cycle domains, the positions of the limit cycles are pointed out accurately. The research way is different from other authors'.In Chapter 4 and Chapter 5, complete researches have been done respectively on the center conditions and center integral of a quintic system and a septic system. Bifurcation of limit cycles from the equator of these systems is studied as well.In Chapter 6, the following general real planar odd polynomial system is studied.where Xi and Yi are homogeneous polynomials of degree i. Two applicable linear algebraicrecursion formulas are given for the quantities of equator of system (2).In the last chapter, by introducing the isochronous center of real systems into complex planar and defining complex center and complex isochronous center, a concise linear recursion formula for period constants is given, necessary and sufficient conditions of complex isochronous center (the Time-Angle Difference Theorem) proved, conditions of real systems with linearizable center and saddle treated unitedly and the isochronous center conditions discussed fully for a class of real planar cubic systems. At last, the isochronous center of infinity of a class of quintic systems is studied and all conditions of center and isochronous center are obtai...