الفهرس | Only 14 pages are availabe for public view |
Abstract One of the most important topics in differential equations is the study of bifurcation theory. This thesis is directed toward the study of local static bifurcation theory using tools from functional analysis. The bifurcation of differential equation is concerned with changes in the qualitative behaviour of its phase portrait as a parameter (or a set of parameters) varies. This thesis contains four chapters and an appendix. The first chapter is considered as an introduction. It contains the basic definitions in connection with the theory of bifurcations with some simple examples. Chapter two introduces the basic tools needed for handling our problem such as basic theorems and methods from analysis and algebra. Chapter three treats the local static bifurcations of differential equations where the dimension of the nullity of the matrix representing the linearized part of the function M (A.~x) (the differential equation under consideration is the equation dxldt-M(A,x) where x is a vector function representing the dependent variable and t is a .real variable representing the independent variable and A. is a vector representing the parameter in the equation) is one or more. In this chapter we make an investigation concerning quadratic and cubic nonlinearities only. Chapter four gives an application in chemical reactions of a model treated in three stages. The first stage of this model was considered by lefever and Nicolis [3]. The second stage of this model was considered by Boa and Cohen [5]. The third stage of this model is a modification of the second stage and we denote it by modified Boa and cohen model. Appendix A gives comparison between bifurcations with one and two dimensional null spaces where we considered our model of chemical reactions as an example on two dimensional null space. |