The z-transform, difference equations, and discrete systems

2018 ◽  
pp. 401-476
Author(s):  
Alexander D. Poularikas
Keyword(s):  
Difference Equations ◽  
Discrete Systems

scholarly journals Direct linearization approach to discrete integrable systems associated with ℤ 𝒩 graded Lax pairs

2020 ◽  
Vol 476 (2237) ◽  
pp. 20200036
Author(s):  
Wei Fu
Keyword(s):  
Difference Equations ◽  
Bilinear Form ◽  
Integrable Systems ◽  
Solution Structure ◽  
Discrete Systems ◽  
Lax Pairs ◽  
Discrete Integrable Systems ◽  
Direct Linearization ◽  
Linear Integral ◽  
Linear Integral Equations

Fordy and Xenitidis [ J. Phys. A: Math. Theor. 50 (2017) 165205. ( doi:10.1088/1751-8121/aa639a )] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of Z N graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy–Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the FX integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the FX novel models and the discrete Gel’fand–Dikii hierarchy.

Existence of periodic solutions with minimal period for fourth-order discrete systems via variational methods

2020 ◽  
Vol 21 (6) ◽  
pp. 635-640
Author(s):  
Lianwu Yang
Keyword(s):  
Periodic Solution ◽  
Variational Method ◽  
Fourth Order ◽  
Difference Equations ◽  
Discrete Systems ◽  
Point Theory ◽  
Existence Results ◽  
Minimal Period ◽  
Nonlinear Difference Equations ◽  
Existence Of Periodic Solutions

AbstractBy using critical point theory, some new existence results of at least one periodic solution with minimal period pM for fourth-order nonlinear difference equations are obtained. Our approach used in this paper is a variational method.

scholarly journals INVARIANT MANIFOLDS FOR COMPETITIVE DISCRETE SYSTEMS IN THE PLANE

2010 ◽  
Vol 20 (08) ◽  
pp. 2471-2486 ◽  
Author(s):  
M. R. S. KULENOVIĆ ◽  
ORLANDO MERINO
Keyword(s):  
Difference Equations ◽  
Invariant Manifolds ◽  
Equilibrium Points ◽  
Global Bifurcation ◽  
Discrete Systems ◽  
Basins Of Attraction ◽  
Competitive Systems ◽  
Invariant Regions ◽  
Competitive Map ◽  
Hyperbolic Equilibrium

Let T be a competitive map on a rectangular region [Formula: see text], and assume T is C1 in a neighborhood of a fixed point [Formula: see text]. The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from [Formula: see text] when both eigenvalues of the Jacobian of T at [Formula: see text] are nonzero and at least one of them has absolute value less than one, and establish that [Formula: see text] is an increasing curve that separates [Formula: see text] into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. These results, known in hyperbolic case, have been used to determine the basins of attraction of hyperbolic equilibrium points and to establish certain global bifurcation results when switching from competitive coexistence to competitive exclusion. The emphasis in applications in this paper is on planar systems of difference equations with nonhyperbolic equilibria, where we establish a precise description of the basins of attraction of finite or infinite number of equilibrium points.

scholarly journals Periodicity in distribution. I. Discrete systems

2002 ◽  
Vol 30 (2) ◽  
pp. 65-127 ◽  
Author(s):  
A. Ya. Dorogovtsev
Keyword(s):  
Banach Space ◽  
Difference Equations ◽  
Discrete Systems ◽  
Solution Stability ◽  
Linear Difference Equations ◽  
Finite Dimensional ◽  
The Cauchy Problem ◽  
Distribution Process ◽  
Dynamic Stochastic ◽  
The Difference

We consider the existence of periodic in distribution solutions to the difference equations in a Banach space. A random process is called periodic in distribution if all its finite-dimensional distributions are periodic with respect to shift of time with one period. Only averaged characteristics of a periodic process are periodic functions. The notion of the periodic in distribution process gave adequate description for many dynamic stochastic models in applications, in which dynamics of a system is obviously nonstationary. For example, the processes describing seasonal fluctuations, rotation under impact of daily changes, and so forth belong to this type. By now, a considerable number of mathematical papers has been devoted to periodic and almost periodic in distribution stochastic processes. We give a survey of the theory for certain classes of the linear difference equations in a Banach space. A feature of our treatment is the analysis of the solutions on the whole of axis. Such an analysis gives simple answers to the questions about solution stability of the Cauchy problem on+∞, solution stability of analogous problem on−∞, or of existence solution for boundary value problem and other questions about global behaviour of solutions. Examples are considered, and references to applications are given in remarks to appropriate theorems.

scholarly journals Exponential Stability of Linear Discrete Systems with Multiple Delays

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
J. Baštinec ◽  
H. Demchenko ◽  
J. Diblík ◽  
D. Ya. Khusainov
Keyword(s):  
Linear System ◽  
Exponential Stability ◽  
Difference Equations ◽  
Lyapunov Method ◽  
Discrete Systems ◽  
Exponential Estimate ◽  
Multiple Delays ◽  
System Of Difference Equations ◽  
New Criterion ◽  
Linear Discrete Systems

The paper investigates the exponential stability and exponential estimate of the norms of solutions to a linear system of difference equations with multiple delays xk+1=Axk+∑i=1sBixk-mi, k=0,1,…, where s∈N, A and Bi are square matrices, and mi∈N. New criterion for exponential stability is proved by the Lyapunov method. An estimate of the norm of solutions is given as well and relations to the well-known results are discussed.

scholarly journals Dynamic Characteristics of Four Systems of Difference Equations with Higher Order

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Abdul Qadeer Khan ◽  
Shahid Mahmood Qureshi ◽  
Imtiaz Ahmed
Keyword(s):  
Fixed Point ◽  
Rate Of Convergence ◽  
Difference Equations ◽  
Dynamic Characteristics ◽  
Discrete Systems ◽  
Higher Order ◽  
Dynamical Characteristics ◽  
Globally Stable ◽  
Theoretical Results

In this paper, we explore the global dynamical characteristics, boundedness, and rate of convergence of certain higher-order discrete systems of difference equations. More precisely, it is proved that for all involved respective parameters, discrete systems have a trivial fixed point. We have studied local and global dynamical characteristics at trivial fixed point and proved that trivial fixed point of the discrete systems is globally stable under respective definite parametric conditions. We have also studied boundedness and rate of convergence for under consideration discrete systems. Finally, theoretical results are confirmed numerically. Our findings in this paper are considerably extended and improve existing results in the literature.

scholarly journals On Dynamics of a Fractional-Order Discrete System with Only One Nonlinear Term and without Fixed Points

Electronics ◽  
2020 ◽  
Vol 9 (12) ◽  
pp. 2179
Author(s):  
Amina-Aicha Khennaoui ◽  
Adel Ouannas ◽  
Shaher Momani ◽  
Iqbal M. Batiha ◽  
Zohir Dibi ◽  
...  
Keyword(s):  
Fixed Points ◽  
Fractional Order ◽  
Difference Equations ◽  
Discrete System ◽  
Nonlinear Term ◽  
Approximate Entropy ◽  
Discrete Systems ◽  
Largest Lyapunov Exponent ◽  
Hidden Attractors ◽  
Chaotic Modes

Dynamical systems described by fractional-order difference equations have only been recently introduced inthe literature. Referring to chaotic phenomena, the type of the so-called “self-excited attractors” has been so far highlighted among different types of attractors by several recently presented fractional-order discrete systems. Quite the opposite, the type of the so-called “hidden attractors”, which can be characteristically revealed through exploring the same aforementioned systems, is almost unexplored in the literature. In view of those considerations, the present work proposes a novel 3D chaotic discrete system able to generate hidden attractors for some fractional-order values formulated for difference equations. The map, which is characterized by the absence of fixed points, contains only one nonlinear term in its dynamic equations. An appearance of hidden attractors in their chaotic modes is confirmed through performing some computations related to the 0–1 test, largest Lyapunov exponent, approximate entropy, and the bifurcation diagrams. Finally, a new robust control law of one-dimension is conceived for stabilizing the newly established 3D fractional-order discrete system.

scholarly journals On asymptotic behavior of solutions of discrete systems

2009 ◽  
Vol 43 (1) ◽  
pp. 99-108
Author(s):  
Irena Hlavičková
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Discrete Systems ◽  
Asymptotic Behavior Of Solutions ◽  
Discrete Equations ◽  
Nonlinear Anal ◽  
The Given

Abstract This contribution concerns the asymptotic behavior of solutions of systems of difference equations. It reassumes paper [J. Dibl´ık: Anti-Lyapunovmethod for systems of discrete equations, Nonlinear Anal. 57 (2004), 1043-1057], where conditions are described which ensure that the graph of at least one solution of the studied system stays in a prescribed domain. Those conditions include the existence of a so called connecting function for the sets forming the given domain. The connecting function must have certain properties which enable the use of the retract method for the proof of the main result of the cited work. We show that the conditions can be made simpler, without the need to introduce the connecting function.

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