Periodic Solution Bifurcation With Two - Parameter Periodic System

The purpose of this paper is to investigate the bifurcations on a class of one-dimensional periodic systems with a small parameter after perturbations. According to the first order analysis of the periodic solutions of periodic differential system with a small parameter, we introduce another small parameter in the right hand of the differential equations and give the averaging theory of first order of the periodic systems with multiple parameters. Furthermore, by studying the expansions of the averaging functions in a small parameter, we discuss the bounds of the number of limit cycles which bifurcate from the periodic orbits of the unperturbed systems.This eaasy is composed of the following three chapters.In Chapter 1, we briefly introduce the research background, preliminary, the main results and the method used in this paper.In Chapter 2, we consider an one-dimensional analytic differential equation with mul-tiple small parameters ε and λ of the form where 0<ε<<λ<<1, G and F are periodic functions with respect to 9. For ε=0, the unperturbed system has a family of periodic orbits for sufficiently small parameter λ. Using some known results related to the periodic system with a small parameter, we give the expression of the first order averaging function M(z, λ). Then by studying the expansions of the averaging function M(z,λ) in λ, we derive the corresponding explicit expressions and obtain the relationship between them and the bound of the number of limit cycles which bifurcate from the periodic orbits of the unperturbed differential equation.In Chapter 3, as application examples, we investigate two families of periodic systems with multiple small perturbations. The first one is a planar polynomial differential system and the second one is an ordinary analytic differential equation with multiple small pa-rameters. By taking proper transformations, the systems become the forms of the system that we have studied in chapter 2. Then we apply the obtained results in Chapter 2 to derive the coefficients in the expanding expressions of the bifurcation functions in a small parameter. Utilizing the relationship between the simple zeros of the coefficients and limit cycles, we can obtain the upper bounds of the number of limit cycles. In the end we give out and study a special example.