scholarly journals On Sufficient Conditions for Chaotic Behavior of Multidimensional Discrete Time Dynamical System

2021 ◽  
Author(s):  
Nor Syahmina Kamarudin ◽  
Syahida Che Dzul-Kifli
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Discrete Dynamical System ◽  
Sufficient Conditions ◽  
Discrete Dynamical Systems ◽  
Continuous Map ◽  
Discrete Time Dynamical System ◽  
Periodicity Property

AbstractIn this work, we look at the extension of classical discrete dynamical system to multidimensional discrete-time dynamical system by characterizing chaos notions on $${\mathbb {Z}}^d$$ Z d -action. The $${\mathbb {Z}}^d$$ Z d -action on a space X has been defined in a very general manner, and therefore we introduce a $${\mathbb {Z}}^d$$ Z d -action on X which is induced by a continuous map, $$f:{\mathbb {Z}}\times X \rightarrow X$$ f : Z × X → X and denotes it as $$T_f:{\mathbb {Z}}^d \times X \rightarrow X$$ T f : Z d × X → X . Basically, we wish to relate the behavior of origin discrete dynamical systems (X, f) and its induced multidimensional discrete-time $$(X,T_f)$$ ( X , T f ) . The chaotic behaviors that we emphasized are the transitivity and dense periodicity property. Analogues to these chaos notions, we consider k-type transitivity and k-type dense periodicity property in the multidimensional discrete-time dynamical system. In the process, we obtain some conditions on $$(X,T_f)$$ ( X , T f ) under which the chaotic behavior of $$(X,T_f)$$ ( X , T f ) is inherited from the original dynamical system (X, f). The conditions varies whenever f is open, totally transitive or mixing. Some examples are given to illustrate these conditions.

scholarly journals Surjunctivity and topological rigidity of algebraic dynamical systems

2017 ◽  
Vol 39 (3) ◽  
pp. 604-619 ◽  
Author(s):  
SIDDHARTHA BHATTACHARYA ◽  
TULLIO CECCHERINI-SILBERSTEIN ◽  
MICHEL COORNAERT
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Countable Group ◽  
Continuous Group ◽  
Continuous Map ◽  
Algebraic Dynamical Systems ◽  
Metrizable Group ◽  
Group Automorphisms

Let$X$be a compact metrizable group and let$\unicode[STIX]{x1D6E4}$be a countable group acting on$X$by continuous group automorphisms. We give sufficient conditions under which the dynamical system$(X,\unicode[STIX]{x1D6E4})$is surjunctive, i.e. every injective continuous map$\unicode[STIX]{x1D70F}:X\rightarrow X$commuting with the action of$\unicode[STIX]{x1D6E4}$is surjective.

scholarly journals Discrete Dynamical Systems: A Brief Survey

2018 ◽  
Vol 14 (1) ◽  
pp. 35-51
Author(s):  
Sara Fernandes ◽  
Carlos Ramos ◽  
Gyan Bahadur Thapa ◽  
Luís Lopes ◽  
Clara Grácio
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Complex Numbers ◽  
Survey Paper ◽  
Number Systems ◽  
Mathematical Formalization ◽  
Research Students ◽  
Work Done

 Dynamical system is a mathematical formalization for any fixed rule that is described in time dependent fashion. The time can be measured by either of the number systems - integers, real numbers, complex numbers. A discrete dynamical system is a dynamical system whose state evolves over a state space in discrete time steps according to a fixed rule. This brief survey paper is concerned with the part of the work done by José Sousa Ramos [2] and some of his research students. We present the general theory of discrete dynamical systems and present results from applications to geometry, graph theory and synchronization. Journal of the Institute of Engineering, 2018, 14(1): 35-51

Approximating Lyapunov Exponents of Discrete Dynamical Systems

2019 ◽  
Vol 11 (11) ◽  
pp. 1612-1615
Author(s):  
Wadia Faid Hassan Al-Shameri
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Science And Technology ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Significant Part ◽  
Matlab Code

Lyapunov exponents play a significant part in revealing and quantifying chaos, which occurs in many areas of science and technology. The purpose of this study was to approximate the Lyapunov exponents for discrete dynamical systems and to present it as a quantifier for inferring and detecting the existence of chaos in those discrete dynamical systems. Finally, the approximation of the Lyapunov exponents for the discrete dynamical system was implemented using the Matlab code listed in the Appendix.

scholarly journals Unfoldings of discrete dynamical systems

1984 ◽  
Vol 4 (3) ◽  
pp. 421-486 ◽  
Author(s):  
Joel W. Robbin
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conjugacy Class ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Topological Conjugacy ◽  
Universal Unfolding ◽  
Moser Iteration

AbstractA universal unfolding of a discrete dynamical system f0 is a manifold F of dynamical systems such that each system g sufficiently near f0 is topologically conjugate to an element f of F with the conjugacy φ and the element f depending continuously on f0. An infinitesimally universal unfolding of f0 is (roughly speaking) a manifold F transversal to the topological conjugacy class of f0. Using Nash-Moser iteration we show infinitesimally universal unfoldings are universal and (in part II) give a class of examples relating to moduli of stability introduced by Palis and De Melo.

Bekenstein–Hawking entropy for discrete dynamical systems on sites

2019 ◽  
Vol 34 (32) ◽  
pp. 1950265
Author(s):  
Sh. Najmizadeh ◽  
M. Toomanian ◽  
M. R. Molaei ◽  
T. Nasirzade
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Upper Bound ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
New Class ◽  
A Site

In this paper, we extend the notion of Bekenstein–Hawking entropy for a cover of a site. We deduce a new class of discrete dynamical system on a site and we introduce the Bekenstein–Hawking entropy for each member of it. We present an upper bound for the Bekenstein–Hawking entropy of the iterations of a dynamical system. We define a conjugate relation on the set of dynamical systems on a site and we prove that the Bekenstein–Hawking entropy preserves under this relation. We also prove that the twistor correspondence preserves the Bekenstein–Hawking entropy.

scholarly journals Repelling invariant curves in planar discrete dynamical systems

1994 ◽  
Vol 49 (3) ◽  
pp. 469-481 ◽  
Author(s):  
Francisco Esquembre
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Continuous Dependence ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Invariant Curves ◽  
Dynamical Properties ◽  
Attracting Fixed Point

Constructive, simple proofs for the existence, regularity, continuous dependence and dynamical properties of a repelling invariant curve for a discrete dynamical system of the plane with an attracting fixed point with real eigenvalues are given. These proofs can be used to generate a numerical algorithm to find these curves and to compute explicitly the dependence of the curve with respect to the system.

scholarly journals On Nonautonomous Discrete Dynamical Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Dhaval Thakkar ◽  
Ruchi Das
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Space ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Topological Stability

We define and study expansiveness, shadowing, and topological stability for a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a metric space.

Attracting invariant curves in planar discrete dynamical systems

1995 ◽  
Vol 51 (2) ◽  
pp. 273-286
Author(s):  
Francisco Esquembre
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Dynamic Behaviour ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Invariant Curves ◽  
Attracting Fixed Point

We study the properties of an invariant attracting curve passing through an attracting fixed point of a planar discrete dynamical system. We compare these properties to the corresponding properties of the invariant repelling curve studied in [3] in order to determine the dynamic behaviour of the system near the fixed point.

First Integrals of Discrete System Based on the Principle of Jourdain

2012 ◽  
Vol 479-481 ◽  
pp. 711-714 ◽  
Author(s):  
Ning Zhang ◽  
Gang Ling Zhao
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Discrete System ◽  
Discrete Dynamical System ◽  
Variational Formula ◽  
Discrete Analog ◽  
Discrete Dynamical Systems ◽  
First Integrals ◽  
Discrete Transformation

In this paper, we investigate first integrals of discrete dynamical systems with the variational principle of Jourdain. The operators of discrete transformation are introduced for the system. Based on the Jourdainian generalized variational formula, we derive the discrete analog of Noether-type identity, and then we obtain the first integrals of discrete dynamical system. We discuss an example to illustrate these results.

A FAMILY OF COMPLETELY PERIODIC QUADRATIC DISCRETE DYNAMICAL SYSTEM

2008 ◽  
Vol 18 (05) ◽  
pp. 1425-1433 ◽  
Author(s):  
MILAN KUTNJAK ◽  
MATEJ MENCINGER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Dynamical System ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Discrete Dynamical Systems ◽  
Continuous Systems ◽  
Nonassociative Algebras ◽  
Rank 2 ◽  
The Basin Of Attraction

There is a one-to-one correspondence between homogeneous quadratic dynamical systems and commutative (possibly nonassociative) algebras. The corresponding theory for continuous systems is well known (c.f. [Markus, 1960; Walcher, 1991; Kinyon & Sagle, 1995]). In this paper the dynamics on the boundary of the basin of attraction of the origin, ∂ B Att (0), in homogeneous quadratic discrete dynamical systems is considered. In particular, we consider the dynamical behavior in a family of systems corresponding to a family of algebras [Formula: see text] which admits nilpotents of rank 2 and idempotents. The complete periodicity of a system (and the corresponding algebra) is defined and it is proven that for every n > 2 there are some systems/algebras from [Formula: see text] which are on ∂ BAtt(0) completely periodic with period n. The dynamics on ∂ B Att (0) is considered via a special class of polynomials Pn, n ∈ ℕ ∪ {0, -1}, recursively defined by Pn(α) = 2αPn-2(α) + Pn-1(α); P-1(α) = 0, P0(α) = 1, n ∈ ℕ.

Sign in / Sign up

Export Citation Format

Share Document