A Class Of Nonlinear Evolution Equations Qualitative State

Qualitative mode of Nonlinear Evolution Equations will be discussed in this thesis, as well as the existence of small amplitude period solutions and the approximate expression of the analytic solution of Hopf bifurcation, at the same time, give the distinction of stability domain in the different parameter plane.In recent years, lots of Differential Equation modes with time delay have been brought out from the fields of building structure, circuit, optics, social economics, environment, medical science, neural network, and mechanics, etc, and important objectives have been obtained. Meanwhile, to control dynamical system by time delay has also been adopted, for example, time delay feed back control has been one of the main methods to deal with chaos. It is of great importance for us understanding such kind of mode to do research in the future.In strict sense, time delay is widespread, so it is especially important to research such phenomenon, which will occupy a decisive position in science development theoretically and practically. To conduct research on time delay is meaningful and challenging. For one reason, time delay greatly influences kinetic property of the system, and often results in the loss of its stability. For another reason, time delay system usually has infinite characteristic values, so we can say it is infinite dimensional infinite dimensional, which makes the research on it much challenging. Now, the research on time delay system concentrates on stability, Hopf bifurcation and chaos, etc. Generally speaking, time delay leads to the whole system losing its stability, thus producing many branches. In the branches of nonlinear time delay dynamical system, the most commonly discussed is Hopf bifurcation, the evolution and occurrence of homoclinic orbits and heterolinic orbits.This thesis falls into five chapters. Chapter one is introduction, which will briefly introduce the background of time delay equation, its recent development and the main topics.Chapter two will explain the dynamic behavior by the method of multiple scales with the common nonlinear vibration system x= f(x,x,t,t-τ,ε) as example.Chapter three will combine and illustrate Iooss and Yoshihisa's Hopf bifurcation.Chapter four and Chapter five will research time delay vibration system, whose mode can be used to describe motion equation of vertical vibration when the buildings and bridges are influenced by horizontal earthquake effects. Chapter four will firstly introduce the stability dividing of such kind of system in parameter level (τ,α), and will then give its approximate expression of light vibration periodic solution with multidimensional analysis method, and will finally judge its stability. Chapter five will work out Hopf bifurcation of such system by means of Hopf bifurcation theory that we have discussed in Chapter three, and will show the different branches under various situations. This analysis result plays an important reference value in revealing the spread of the earthquake wave and its damage effect.