scholarly journals Stability of a time discrete perturbed dynamical system with delay

1999 ◽  
Vol 3 (1) ◽  
pp. 57-63 ◽  
Author(s):  
Michael I. Gil' ◽  
Sui Sun Cheng
Keyword(s):  
Dynamical System ◽  
Positive Integer ◽  
Difference Equation ◽  
Absolute Stability ◽  
Nonlinear Difference Equation ◽  
Stability Conditions ◽  
System With Delay ◽  
Equation With Delay

LetCnbe the set ofncomplex vectors endowed with a norm‖⋅‖Cn. LetA,Bbe two complexn×nmatrices andτa positive integer. In the present paper we consider the nonlinear difference equation with delay of the typeuk+1=Auk+Buk−τ+Fk(uk,uk−τ),      k=0,1,2,…, whereFk:Cn×Cn→Cnsatisfies the condition‖Fk(x,y)‖Cn≤p‖x‖Cn+q‖y‖Cn,       k=0,1,2,…, wherepandqare positive constants. In this paper, absolute stability conditions for this equation are established.

Asymptotic stability of fractional difference equations with bounded time delays

2020 ◽  
Vol 23 (2) ◽  
pp. 571-590
Author(s):  
Mei Wang ◽  
Baoguo Jia ◽  
Feifei Du ◽  
Xiang Liu
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Integral Inequality ◽  
Time Delays ◽  
Asymptotical Stability ◽  
Stability Conditions ◽  
Fractional Difference ◽  
Fractional Difference Equations ◽  
Fractional Difference Equation

AbstractIn this paper, an integral inequality and the fractional Halanay inequalities with bounded time delays in fractional difference are investigated. By these inequalities, the asymptotical stability conditions of Caputo and Riemann-Liouville fractional difference equation with bounded time delays are obtained. Several examples are presented to illustrate the results.

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.

scholarly journals Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
G. M. Moremedi ◽  
I. P. Stavroulakis
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
First Order ◽  
All Solutions ◽  
Delay Difference ◽  
Constant Argument

Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0,  n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0,  n=0,1,2,…, where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

scholarly journals ON EXISTENCE OF POSITIVE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 257-265
Author(s):  
J. H. SHEN ◽  
Z. C. WANG ◽  
X. Z. QIAN
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Existence Of Positive Solutions

Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.

A Class of Hyperspace Systems and Distributional Chaos

2014 ◽  
Vol 670-671 ◽  
pp. 1570-1572
Author(s):  
Wei Wang ◽  
Xiao Gang Zhu
Keyword(s):  
Dynamical System ◽  
Difference Equation ◽  
Agricultural Production ◽  
Irrigation District ◽  
Corn Yield ◽  
Primitive Substitution ◽  
Yield Analysis ◽  
Distributional Chaos ◽  
Dynamical Properties ◽  
Ecological Difference

Research on dynamical system has penetrated into many problems of agricultural production, such as prediction of corn yield, analysis on operational situation of irrigation district and research on ecological difference equation. In this paper, we investigated dynamical properties for non-primitive substitution and the set-valued maps induced by the substitution. We proved In two cases that the hyperspace systems induced by non-primitive substitution are not distributional chaotic.

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