Affine-Periodic Solutions Of Differential Equation

Cycle of movement is a universal phenomenon in nature. The growth of crops?periodic oscillation caused by the electron transport the diurnal cycle in the light period?the celestial phenomena between universe of stars go round and begin again, and so on,all these reveal the periodic motion. We use differential equations as a tool to characterize these phenomena, the solution of differential equations, we should reflect a certain periodicity in time. As a form of expression of the beauty of nature, differential equation often show a certain symmetry, not just a cyclical, for example, the affine symmetry:Consider the following system whereQ ? GL(Rn). Then they are called(Q, ?)-Affine cycle system. This paper mainly studies the problem of affine periodic on differential equations using the "guiding function method" by Mawhin, that is, if a nonlinear system has a certain symmetry, then in a certain transversality condition and a priori estimate, this system has corresponding affine periodic solution.This dissertation is divided by three chapters.In the first chapter, we mainly introduce a brief history of the development of differential equation, problems related to periodic solutions of ordinary dif-ferential equations, review some classical results about periodic solution, such as:Krasnoselskii theorem, Mawhin theorem, and give the Mawhin's "guiding function method".In the second chapter, we discuss the rotating-symmetric solutions for nonlinear systems with symmetry. Consider the equation: that is to say f is T-periodic in t the problem that people cares is the existence of T-periodic solutions for equation. Mawhin[33]propose Mawhin's Theorem, the result is generalized to the existence of periodic solutions for impulsive equations by Li-Yong[60] et al.In this paper, we discuss:for certain^ S SO(n), research the rotating-symmetric solutions of x= f(t,x) by "guiding function method".Assume that f(t+T,x)= Qf(t,Q-1(x))is true forall(t,x) and there exist C1 functions Vi(x),i= 0,1,2, …,m, such that:(1)for Mi large enough,Then (0.0.4)has Q-rotating symmetric solutionsx (t), i.e., x(t+T)=Qx(t) Vt.In the third chapter, We introduce the LaSalle stationary oscillation the-orem. First, we prove the similar stationary oscillation theorem to Q-affine periodic system, and discuss a more general stationary oscillation condition, give more convenient application form,(1)the solutions z(t) of (0.0.4) defined on R+1;(2)Any two solutions x(t, x0)?x(t,y0)satisfies then (0.0.4)has a unique asymptotic stability Q-rotating T-periodic solution.Next, we prove that x= f(t, x) has a unique asymptotic stability Q-rotating T-periodic solution, if it is asymptotic stability.In the end, We discuss the existence of Q-rotating T-periodic solution and asymptotic stability of the system x= F(t,xt), that is to have the cor-responding results of ordinary differential equations to functional differential equation with delay limit.