Bifurcation Analysis In Some Kinds Of Delay Differerntial Equations

Since the time when Newton and Leibniz established calculus, people have be-gun to use differential equations to describe the evolution systems in the objectiveworld, where bifurcation problem is one of the best important subjects in dynamicalsystem and nonlinear differential equations. The main research project of bifurca-tion theory is to investigate such phenomenon in the structural unstable systems thatthe topological structures make changes provided the parameters in the systems varywhen they cross certain critical values. Bifurcation problem includes local bifurca-tion, semi-local bifurcation and global bifurcation. Hopf bifurcation, a kind of localbifurcation, studies the changes of the stability of the equilibrium point, when theparameters in the system are varied, that is, considers the occurrence of the smallamplitude oscillatory periodic solutions near the equilibrium point.Retard differential equations (RDEs) refers to the differential equations withtime delays, which can be used to describe the evolution systems dependent on boththe present state and the past state. Since the in?uences of history on the present stateare sufficiently considered, RDEs has wide applications in ecology, physics, chem-istry, engineering, information science, economics and physical science and so on.The investigation of the bifurcation problem in RDEs, needs not only the theory ofthe classical dynamical system and the differential equations, but also the knowledgeof functional, geometry, topology one. Hence, the systematic and deep investigationof the bifurcation theory is of great significance in both intense practical and theoreti-cal background.This paper deals with bifurcation for some kinds of evolution systems with delay,and the main theoretical structure of the paper is stated as follows.Taking delayτas the bifurcation parameter, we derive the conditions for theequilibrium point to lose its stability whenτcrosses zero and becomes larger gradu-ally. The main way is to discuss the distribution of the roots of characteristic equationfor the linearized system, where the comprehensive and extensively applied methodsand theory due to Junjie Wei and Shigui Ruan are used. Based on the center manifoldtheory of abstract differential equations, we calculate the bifurcation direction and the stability of the bifurcating periodic solutions in the following way: Firstly, by project-ing the retarded functional differential equation into the center manifold and utilizingthe normal form methods, we can derive the conditions to determine the bifurcationdirection, the stability, amplitude, period of the bifurcating periodic solutions, and theformula of the projection in the center manifold. Based on the Hopf bifurcation prop-erties computed by Kazarinoff, Junjie Wei and Shigui Ruan were able to generalizean effective formula to calculate the properties of Hopf bifurcation for FDEs. Withthis formula, in order to obtain more detailed information on Hopf bifurcation prop-erties, one need only to substitute the known variables related to the system to theformula. Now another problem arises: whether the bifurcating periodic solutions stillhold, when the parameterτcrosses the first bifurcation value and becomes larger andlarger continuously? That is to say, whether the so-called global Hopf bifurcating pe-riodic solutions exist? To solve this problem, we use the Bendixson criterion of highdimension due to M.Li et al, and the existence theorem of global periodic solutionsdue to Jianhong Wu, together with the technics summarized by Junjie Wei and M. Li.We derive the sufficient conditions for the bifurcating periodic solutions to occur inthe large scale of the parameter, and determine the lower limit of the number of thepossible global bifurcating periodic solutions.In 1988,M. Golubitsky et al. published their well-known book, entitled"Singu-larities and groups in bifurcation theory". Based on the equivariant bifurcation theory,they revealed the fact that the circle composed by the exact same cells could give riseto some singular and interesting oscillations. Later, Jianhong Wu et al. extended suchequivariant Hopf bifurcation theory to DDEs, which tends to be an effective method todeal with the case when the pure imaginary characteristic roots are not simple . Theirwork is the theoretic foundation for studying neural network of this structure.Based on the instruction of the above theories, the main creativities in the thesisare as follows:1. From the viewpoint of bifurcation, we study the well-known physical modelproposed by Mackey and Glass. We derive the conditions for the system to undergoa Hopf bifurcation near the unique positive equilibrium point when the delayτin-creases. Then by utilizing the theory due to B.D. Hassard and K.D. Kazarinoff et al,and the method due to Junjie Wei and Shigui Ruan, we obtain the formula to deter-mine the bifurcation direction, bifurcation periodic solutions, amplitude and period, and the conditions for the global periodic solutions to occur are investigated at the endof this part.2. We study the Hopf bifurcation problem of the neural network of three-neuralelement with delay. By analyzing the characteristic roots, we show the stability of thezero equilibrium point, and derive the conditions for the Hopf bifurcation to occur.Secondly, by using center manifold theorem and normal form method, we derive theformula to determine the Hopf bifurcation direction and stability of bifurcation peri-odic solutions. And then, we perform detailed global Hopf bifurcation analysis to thesystem. Finally, with some examples, we use the numerical simulations to support ouranalytical results.3. We consider the Hopf bifurcation of delayed neural network with the symmet-rical structure and n-neural elements. To begin with, we divide our investigation intothree cases: n is odd, n is even but not the integral multiple of four, and the case whenn is the of integral multiple 4. Then we give the detailed expression of characteristicequation of the linearized system accordingly. Following the above three cases, weseparately study the conditions, under which the system might occur Hopf bifurcationnear equilibrium point. And for the conditional stable region, we do detailed partition.Secondly, based on the theoretical instruction due to M. Golubitsky and Jianhong Wu,where the tool of Lie group and Faria's normal form theory are used, we derive thesufficient conditions for the 2(n+1) non-synchronous period three solutions to occur.Moreover, in each chapter, by using the theory due to Faria et al, we provide the de-tailed computational process of the normal form in the center manifold of equilibriumpoint for the symmetric structural system accordingly.