scholarly journals Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex

2016 ◽  
Vol 14 (1) ◽  
pp. 509-519 ◽  
Author(s):  
Rawad Abdulghafor ◽  
Farruh Shahidi ◽  
Akram Zeki ◽  
Sherzod Turaev
Keyword(s):  
Fixed Points ◽  
Extreme Points ◽  
Initial Point ◽  
Higher Dimensions ◽  
Periodic Points ◽  
Two Dimensional ◽  
Invariant Curves ◽  
Dimensional Simplex ◽  
Doubly Stochastic ◽  
Finite Dimensional

Abstract The present paper focuses on the dynamics of doubly stochastic quadratic operators (d.s.q.o) on a finite-dimensional simplex. We prove that if a d.s.q.o. has no periodic points then the trajectory of any initial point inside the simplex is convergent. We show that if d.s.q.o. is not a permutation then it has no periodic points on the interior of the two dimensional (2D) simplex. We also show that this property fails in higher dimensions. In addition, the paper also discusses the dynamics classifications of extreme points of d.s.q.o. on two dimensional simplex. As such, we provide some examples of d.s.q.o. which has a property that the trajectory of any initial point tends to the center of the simplex. We also provide and example of d.s.q.o. that has infinitely many fixed points and has infinitely many invariant curves. We therefore came-up with a number of evidences. Finally, we classify the dynamics of extreme points of d.s.q.o. on 2D simplex.

scholarly journals On the finite-dimensional marginals of shift-invariant measures

2011 ◽  
Vol 32 (5) ◽  
pp. 1485-1500 ◽  
Author(s):  
J.-R. CHAZOTTES ◽  
J.-M. GAMBAUDO ◽  
M. HOCHMAN ◽  
E. UGALDE
Keyword(s):  
Finite Type ◽  
Invariant Measures ◽  
Convex Set ◽  
Extreme Points ◽  
Higher Dimensions ◽  
Finite Alphabet ◽  
Finite Dimensional ◽  
Shift Action ◽  
Unique Invariant Measure ◽  
Dimension One

AbstractLet Σ be a finite alphabet, Ω=Σℤdequipped with the shift action, and ℐ the simplex of shift-invariant measures on Ω. We study the relation between the restriction ℐnof ℐ to the finite cubes {−n,…,n}d⊂ℤd, and the polytope of ‘locally invariant’ measures ℐlocn. We are especially interested in the geometry of the convex set ℐn, which turns out to be strikingly different whend=1 and whend≥2 . A major role is played by shifts of finite type which are naturally identified with faces of ℐn, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of ℐn, although in dimensiond≥2 there are also extreme points which arise in other ways. We show that ℐn=ℐlocnwhend=1 , but in higher dimensions they differ fornlarge enough. We also show that while in dimension one ℐnare polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of ℐnfor all large enoughn.

scholarly journals EDSQ Operator on 2DS and Limit Behavior

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 820 ◽  
Author(s):  
Rawad Abdulghafor ◽  
Hamad Almohamedh ◽  
Badr Almutairi ◽  
Sharyar Wani ◽  
Abdullah Alharbi ◽  
...  
Keyword(s):  
Mathematical Models ◽  
Nonlinear Model ◽  
Limit Behavior ◽  
Two Dimensional ◽  
Quadratic Operator ◽  
Dimensional Simplex ◽  
Doubly Stochastic ◽  
Nonlinear Mathematical Models ◽  
Quadratic Stochastic Operators

This paper evaluates the limit behavior for symmetry interactions networks of set points for nonlinear mathematical models. Nonlinear mathematical models are being increasingly applied to most software and engineering machines. That is because the nonlinear mathematical models have proven to be more efficient in processing and producing results. The greatest challenge facing researchers is to build a new nonlinear model that can be applied to different applications. Quadratic stochastic operators (QSO) constitute such a model that has become the focus of interest and is expected to be applicable in many biological and technical applications. In fact, several QSO classes have been investigated based on certain conditions that can also be applied in other applications such as the Extreme Doubly Stochastic Quadratic Operator (EDSQO). This paper studies the behavior limitations of the existing 222 EDSQ operators on two-dimensional simplex (2DS). The created simulation graph shows the limit behavior for each operator. This limit behavior on 2DS can be classified into convergent, periodic, and fixed.

scholarly journals Dynamical Behavior of Two Dimensional Hénon Maps

2012 ◽  
Vol 47 (1) ◽  
pp. 55-60
Author(s):  
MS Islam ◽  
MS Islam
Keyword(s):  
Fixed Points ◽  
Computer Programming ◽  
Dynamical Behavior ◽  
Periodic Points ◽  
Numerical Results ◽  
Two Dimensional ◽  
Henon Maps ◽  
Hénon Maps ◽  
Non Linear ◽  
Parameter Values

In this article, we study the two dimensional non-linear dynamical behavior of Hénon maps. We investigate the parameter values for which fixed points and periodic points of period two exist and study the dimension of the maps. We also investigate the numerical results of the maps and use computer programming Mathematica for generating graphs and computations. DOI: http://dx.doi.org/10.3329/bjsir.v47i1.10722 Bangladesh J. Sci. Ind. Res. 47(1), 55-60, 2012

scholarly journals An Energy Efficient QAM Modulation with Multidimensional Signal Constellation

2016 ◽  
Vol 62 (2) ◽  
pp. 159-165 ◽  
Author(s):  
Tomasz G. Markiewicz
Keyword(s):  
Energy Efficient ◽  
Average Energy ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
Sphere Packings ◽  
Dimensional Simplex ◽  
Signal Constellation ◽  
Multidimensional Signal ◽  
Qam Modulation ◽  
The Impact

Abstract Packing constellations points in higher dimensions, the concept of multidimensional modulation exploits the idea drawn from geometry for searching dense sphere packings in a given dimension, utilising it to minimise the average energy of the underlying constellations. The following work analyses the impact of spherical shaping of the constellations bound instead of the traditional, hyper-cubical bound. Balanced constellation schemes are obtained with the N-dimensional simplex merging algorithm. The performance of constellations of dimensions 2, 4 and 6 is compared to the performance of QAM modulations of equivalent throughputs in the sense of bits transmitted per complex (two-dimensional) symbols. The considered constellations give an approximately 0:7 dB to 1 dB gain in terms of BER over a standard QAM modulation.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

Discrete-Time Memristor Model for Enhancing Chaotic Complexity and Application in Secure Communication

2022 ◽  
Author(s):  
Wenhao Yan ◽  
Zijing Jiang ◽  
Qun Ding
Keyword(s):  
Fixed Points ◽  
Discrete Time ◽  
Dynamic Characteristics ◽  
Secure Communication ◽  
Chaotic Systems ◽  
Two Dimensional ◽  
Initial State ◽  
One Dimensional ◽  
Backward Euler ◽  
Short Period

Abstract The physical implementation of continuoustime memristor makes it widely used in chaotic circuits, whereas discrete-time memristor has not received much attention. In this paper, the backward-Euler method is used to discretize TiO2 memristor model, and the discretized model also meets the three fingerprinter characteristics of the generalized memristor. The short period phenomenon and uneven output distribution of one-dimensional chaotic systems affect their applications in some fields, so it is necessary to improve the dynamic characteristics of one-dimensional chaotic systems. In this paper, a two-dimensional discrete-time memristor model is obtained by linear coupling the proposed TiO2 memristor model and one-dimensional chaotic systems. Since the two-dimensional model has infinite fixed points, the stability of these fixed points depends on the coupling parameters and the initial state of the discrete TiO2 memristor model. Furthermore, the dynamic characteristics of one-dimensional chaotic systems can be enhanced by the proposed method. Finally, we apply the generated chaotic sequence to secure communication.

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