Bifurcation Analysis For A Nonlinear Recurrence Relation With Periodic Coefficients And Piecewise Continuous Control

Difference equations is an important branch of dynamical system, and it has important meaning and value in the mathematical model establishment. Difference equations are divided into linear and nonlinear difference equations. In the practical model, the application of the nonlinear difference equation is more extensive, and the less model is applied to the constant coefficient difference equation, So the study dy-namical behavior of nonlinear difference equations with variable coefficients, including the periodicity, attract, asymptotic behavior and stability of more practical signifi-cance.This paper studies a class of nonlinear difference systems with periodic coefficients and a piecewise continuous control, according to the coefficient of the cycle and trans-formation skill will system is divided into two kinds of analysis, that is, coefficients are even periodic and odd periodic sequences. For the first case, that is the coefficient is even periodic sequences, we introduction of matrix and vector, through the matrix of the basic algorithm, describe the system in vector form, carries on the analysis, using the recursive analysis method and apagoge to proof the asymptotic behavior of solutions. For the second case, that is the coefficient is odd periodic sequences, use of the original system, using direct analysis method to get the solution of the system of asymptotic conclusions.This paper is divided into five chapters. Chapter one is the introduction part, first of all, Introduces the research background and development status of non-linear difference equations with piecewise control, followed by an overview of the main con-tents of this paper; the second chapter is some related definitions and notations. The third chapter mainly studies the following form of nonlinear difference equation where are 2κperiodic sequences withai∈(0, 1),bi=1-αi,i= 0,1,…,2κ-1. f satisfies Which λ∈(-∞,+∞).By the coeffieient matrix,the 2κ-dimensional system can be obtained. where we make 2κper cycle as a whol,by analysis the above 2κ-dimensional system ob-tained the asymptotic results of solutions of this system..The fourth chapter mainly studies are 2κ+1 periodic sequences with αi∈(0,1),bi=1-αi,i= 0,1,…,2κ,by the transform xn=x(2κ+1)n+i,κ∈Z,i∈0,1,...,2κ,we the above equation can be converted into the folloWing 2κ+1—dimensional autonomous dynam— ical system by analysis the above 2κ+1-dimensional system obtained the asymptotic results of solutions of this system...