Global behavior of nonlinear difference equations
We investigate the global asymptotic stability, the periodic nature, the rate of convergence, and the bounded character of solutions of two rational difference equations and a system of rational equations. ^ We analyze the behavior of solutions of the equation xn+1=gxn-1+dx n-2A+Bxn, n=0,1,&ldots;, 0.1 with non-negative parameters and non-negative initial conditions. This is a special case of the more general equation xn+1=a+bxn+g xn-1+dxn-2A+Bxn +Cxn-1+Dxn-2, n=0,1,&ldots;,0 .2 and is one of the first third order rational difference equations to be investigated in which all three terms are present in the equation. A thorough analysis of a specific period-two pathology of solutions for the case A = 0 is presented. ^ The behavior of solutions of the system xn+1=h+xna+yn ,yn+1=y nb+xn,n=0 ,1,&ldots;, 0.3 with non-negative parameters and non-negative initial conditions is completely characterized for initial conditions in the second and fourth quadrant of the positive equilibrium in the case when neither a nor b is equal to 1. ^ The stability of the Gumovski-Mira equation xn+1=xnb+x2 n-xn-1,n+0,1,&ldots; , 0.4 is shown to be preserved when the coefficient b is a periodic sequence with prime period two. The behavior of solutions for the case where the even or odd subsequence of the coefficient is zero is also presented. ^
Cathy Ann Clark,
"Global behavior of nonlinear difference equations"
Dissertations and Master's Theses (Campus Access).