Qualitative Analysis of Some Higher Order Rational Difference Equations

Abstract

The study of Qualitative Analysis of rational difference equation is very important. However, in Physics, Economics and Biological sciences difference equations are extensively used. Qualitative behavior includes global attractivity, bounded character and periodicity. The motivation is to study the Dynamics of some higher order rational difference equations. In this thesis we have studied the behavior of solutions of some higher order rational difference equations. To confirm the proved results we have used mathematical program Matlab to give graphical examples by assigning different numerical values to initial values.

The first chapter presents brief introduction to the problem of study, its objectives and methodology used to achieve the goal.

In the second chapter we have presented a comprehensive form of literature review related to the problem understudy. The current literature is discussed in detail and results are summarized to draw conclusions and further directions.

The third chapter is preliminaries, showing notions and basic definitions which are associated with this study.

In chapter four, we have presented our main results with their proofs for rational difference equation of order twenty

2 9 19 9 19 , 0,1,2,...nnn nn z z z n zz         

With initial conditions 19 18 17 16 15 14 13 , , , , , , , z z z z z z z        12 11,, zz 

10,z  9 8 7 6 5 4 3 2 1 0 , , , , , , , , , z z z z z z z z z z R           and the coefficients , , ,     are constants. We obtained some special cases of this equation. The numerical verification and comparison of each graph is also included in this chapter.

In chapter five, we have studied the global stability of the positive solutions and periodic character of the difference equation

0 1 2 3 1 0 1 2 4 5 6 7 n t n l n m n p n n n k n s n t n l n m n p b z b z b z b z z z z z b z b z b z b z                     

x

With non-negative initial conditions 1 1 0 , ,..., , z z z z    where   max , , , , , k s t l m p and the coefficients 0 1 2 0 1 , , , , , bb   2 3 4 , , , b b b 5 6 7 ,, b b b R  . Numerical examples are also given to confirm the obtained results.