Global behavior of solutions of a nonlinear difference equation

2004 ◽  
Vol 159 (1) ◽  
pp. 29-35 ◽  
Author(s):  
D.C. Zhang ◽  
B. Shi
Keyword(s):  
Difference Equation ◽  
Nonlinear Difference Equation ◽  
Global Behavior

Global behavior of solutions of the nonlinear difference equation

2005 ◽  
Vol 11 (8) ◽  
pp. 707-719 ◽  
Author(s):  
R. Devault ◽  
V.L. Kocic ¶ ◽  
D. Stutson §
Keyword(s):  
Difference Equation ◽  
Nonlinear Difference Equation ◽  
Global Behavior

scholarly journals On the global behavior of the nonlinear difference equationxn+1=f(pn,xn−m,xn−t(k+1)+1)

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Taixiang Sun ◽  
Hongjian Xi
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Positive Sequence ◽  
Nonlinear Difference Equation ◽  
Global Behavior ◽  
Initial Values

We consider the following nonlinear difference equation:xn+1=f(pn,xn−m,xn−t(k+1)+1),n=0,1,2,…, wherem∈{0,1,2,…}andk,t∈{1,2,…}with0≤m<t(k+1)−1, the initial valuesx−t(k+1)+1,x−t(k+1)+2,…,x0∈(0,+∞), and{pn}n=0∞is a positive sequence of the periodk+1. We give sufficient conditions under which every positive solution of this equation tends to the periodk+1solution.

scholarly journals Global Behavior of a Nonlinear Difference Equation with Applications

2012 ◽  
Vol 02 (02) ◽  
pp. 78-81
Author(s):  
Decun Zhang ◽  
Jie Huang ◽  
Liying Wang ◽  
Wenqiang Ji
Keyword(s):  
Difference Equation ◽  
Nonlinear Difference Equation ◽  
Global Behavior

scholarly journals Global behavior of a higher order nonlinear difference equation

2004 ◽  
Vol 299 (1) ◽  
pp. 113-126 ◽  
Author(s):  
Yonghong Fan ◽  
Linlin Wang ◽  
Wantong Li
Keyword(s):  
Difference Equation ◽  
Higher Order ◽  
Nonlinear Difference Equation ◽  
Global Behavior

scholarly journals Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 825
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Arbitrary Order ◽  
Nonnegative Function ◽  
Nonlinear Difference Equation ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Asymptotic Behaviors

Let k , l be two integers with k ≥ 0 and l ≥ 2 , c a real number greater than or equal to 1, and f a multivariable function satisfying f ( w 1 , w 2 , w 3 , ⋯ , w l ) ≥ 0 when w 1 , w 2 ≥ 0 . We consider an arbitrary order nonlinear difference equation with the indicated function f: z n + 1 = c ( z n + z n − k ) + ( c − 1 ) z n z n − k + c f ( z n , z n − k , w 3 , ⋯ , w l ) z n z n − k + f ( z n , z n − k , w 3 , ⋯ , w l ) + c , n ≥ 0 , where initial values z − k , z − k + 1 , ⋯ , z 0 are positive and w i , i ≥ 3 , are arbitrary functions of z j , n − k ≤ j ≤ n . We classify its solutions into three types with different asymptotic behaviors, and verify the global asymptotic stability of its positive equilibrium solution z ¯ = c .

Behavior of Dynamical Systems Governed by a Simple Nonlinear Difference Equation

1975 ◽  
Vol 42 (4) ◽  
pp. 870-876 ◽  
Author(s):  
C. S. Hsu ◽  
H. C. Yee
Keyword(s):  
Dynamical Systems ◽  
Difference Equation ◽  
System Parameter ◽  
Initial Conditions ◽  
General Pattern ◽  
Nonlinear Difference Equation ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Nonlinear Difference Equations ◽  
Stable Periodic

Many dynamical systems and mechanics problems are governed by nonlinear difference equations. These equations have also been used increasingly to model problems in population dynamics, economics, and ecology. In this paper we study systems governed by the nonlinear difference equation (4). The locally asymptotically stable periodic solutions are investigated and the global behavior of the system for different values of the system parameter and for different initial conditions is examined. Although the equation is a simple one, the general pattern of its solution is surprisingly complex and seems to have implications in many fields.

DRY TURBULENCE FROM A TIME-DELAYED CHUA'S CIRCUIT

1993 ◽  
Vol 03 (02) ◽  
pp. 645-668 ◽  
Author(s):  
A. N. SHARKOVSKY ◽  
YU. MAISTRENKO ◽  
PH. DEREGEL ◽  
L. O. CHUA
Keyword(s):  
Difference Equation ◽  
Stability Region ◽  
Nonlinear Difference Equation ◽  
Chua’S Circuit ◽  
Stability Boundaries ◽  
Chua's Circuit ◽  
Infinite Dimensional ◽  
Chaotic Region ◽  
The Stability ◽  
Left Portion

In this paper, we consider an infinite-dimensional extension of Chua's circuit (Fig. 1) obtained by replacing the left portion of the circuit composed of the capacitance C2 and the inductance L by a lossless transmission line as shown in Fig. 2. As we shall see, if the remaining capacitance C1 is equal to zero, the dynamics of this so-called time-delayed Chua's circuit can be reduced to that of a scalar nonlinear difference equation. After deriving the corresponding 1-D map, it will be possible to determine without any approximation the analytical equation of the stability boundaries of cycles of every period n. Since the stability region is nonempty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the "period-adding" phenomenon where every two consecutive cycles are separated by a chaotic region.

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