الفهرس | Only 14 pages are availabe for public view |
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. |