Bifurcation Of Several Kinds Of Differential Equations

In this dissertation, by using the method of the Poincare maps and the displacement functions, we firstly study the existence, stability and its criterion and bifurcation of periodic solutions of scaler periodic impulsive differential equations. We also study the bifurcation of planar Hamiltonian differential systems under periodic impulsive perturbation. Most results on periodic solution and periodic boundary value problem are obtained by the method of upper and lower solution, average method, and so on. However, we take the method parallel to that of studying periodic solutions and bifurcations of ordinary differential equations. Moreover, we also study one kind of planar Hamiltonian system under polynomial perturbation, and obtain some interested results. Finally, we consider a kind of resonant center, and give the necessary and sufficient conditions for a l:-3 resonant center with homogeneous cubic polynomial to have a center.The main content of this paper is in the following:In Chapter I, we firstly introduce the impulsive system and its background and results we know. Then we simply introduce resonant center problem and some results have been obtained. At last, we list some innovative point in our work.In Chapter 2, we introduce the notion of Poincare map P and the displacement function to first-order periodic impulsive differential equation. By considering the property of function P and the displacement function, we obtain some conditions for the existence of periodic solutions and the criterion of stability of periodic solutions. A the same time, we study perturbed periodic impulsive differential system. We prove that a simple periodic solution is structurally stable, and obtain the result on saddle-node parallel to that of ordinary differential equation. Moreover, we prove that period-doubling bifurcation of periodic solutions, which is impossible without impulsive effects, occurs. Finally, we give some example to illustrate our theory.In Chapter 3, by means of the Melnikov functions we consider bifurcation problem of a periodic solution of a planar Hamiltonian system under periodic im- pulsive perturbation, and obtain some new results with some sufficient conditions under which a subharmonic or harmonic solution exists, which is parallel to that of ordinary differential system without impulsive effect. Finally, we give an example to illustrate that, adding some impulsive perturbation to a perturbed planar Hamiltonian system will maybe bring more subharmonic solutions.Chapter 4 concern on the bifurcation of perturbed planar Hamilton system. However, we consider the perturbation system of a planar Hamiltonian system taking some function with small parameter as its Hamiltonian function. It's equivalent to add perturbation to a perturbed planar Hamiltonian system and obtain some conditions to generate more periodic solutions. Then taking a special planar Hamiltonian system as an example, by Melnikov function we study the property of the period of periodic solutions of corresponding Hamiltonian system.In Chapter 5, we study l:-3 resonant center with homogeneous cubic polynomial and present the sufficient and necessary conditions, under which the system has a center. To prove the sufficiency, we not only construct a first integral by using invariant curves and its cofactor as others did, but also take the method of constructing a polynomial series as a first integral and even a special method to study some special case.