الفهرس 
INTRODUCTION AND STATEMENT OFOnly 14 pages are availabe for public view
 |  | from | 104 |  |  |
| | | from | 104 | | |
Abstract
This work aims at finding the bifurcations which happen in a
second order conservative recurrence. These bifurcations take place
when a parameter ”a” changes in a certain domain making the
behaviour of the recurrence solution undergoes a qualificative
change.
Practical examples on many problems which lead to recurrence
relations are introduced to show the dependence of many engineering
problems on this work.
The basic theory of recurrences, its order and type, the
solution and its stability, nature and type of singularities and
its description in the phase plane , the possible cases of
bifurcations, all have been investigated and discussed in detail by
means of definitions , simple figures, various solved examples and
well studied cases.
The most important part of this work is the problem of Henan
which is considered here as an example of a second order
conservative recurrence. Its fixed points, many cycles, and the
eigenvalues and type of these singularities are obtained
analytically. Also, critical and exceptional cases are discussed,
many types of bifurcations are deduced and many bifurcation schemes
are obtained.
The present thesis cosists of three chapters. Chapter 1 presents a
general introduction to the thesis and gives some problems which
lead to recurrence relations.
Chapter 2 deals with a survey of the different properties of second
order autonomous recurrences of real variables.
In chapter 3 we consider the recurrence relation
which is an example of a conservative recurrence of the second
order and obtain some of its singularities. Critical and
exceptional cases of the recurrence under study are introduced as
well as many bifurcation schemes.
Finally in appendix A we introduce the equation of transverse
motion in an ideal cyclotron and show that it is possible to find
a corresponding second order conservative recurrence which
describes the same motion.