الفهرس 
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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.