Chaotic Dynamical Systems Associated with Tilings of RN

Dimensional Torus ◽

Synchronization Mechanism ◽

Chaotic Dynamical Systems ◽

Regular Tiling

In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of RN, whose most familiar example is provided by the N-dimensional torus TN. It is proved that any dynamical system in this class is chaotic in the sense of Devaney, and that it admits at least one positive Lyapunov exponent. Next, a chaos-synchronization mechanism is introduced and used for masking information in a communication setup.

Chaos synchronization for a class of discrete dynamical systems on the N-dimensional torus

Systems & Control Letters ◽

10.1016/j.sysconle.2005.07.003 ◽

2006 ◽

Vol 55 (3) ◽

pp. 223-231 ◽

Cited By ~ 6

Author(s):

Lionel Rosier ◽

Gilles Millérioux ◽

Gérard Bloch

Keyword(s):

Dynamical Systems ◽

Chaos Synchronization ◽

Dimensional Torus

NONLINEAR DYNAMICAL SYSTEM AND CHAOS SYNCHRONIZATION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408021142 ◽

2008 ◽

Vol 18 (05) ◽

pp. 1531-1537 ◽

Cited By ~ 3

Author(s):

AYUB KHAN ◽

PREMPAL SINGH

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Chaos Synchronization ◽

Nonlinear Dynamical Systems ◽

Nonlinear Dynamical System ◽

Numerical Techniques ◽

Nonlinear Dynamical ◽

Chaotic Dynamical Systems ◽

Response System ◽

Identical Response

Chaos synchronization of nonlinear dynamical systems has been studied through theoretical and numerical techniques. For the synchronization of two identical nonlinear chaotic dynamical systems a theorem has been constructed based on the Lyapunov function, which requires a minimal knowledge of system's structure to synchronize with an identical response system. Numerical illustrations have been provided to verify the theorem.

On the Lyapunov Exponent of Monotone Boolean Networks †

Mathematics ◽

10.3390/math8061035 ◽

2020 ◽

Vol 8 (6) ◽

pp. 1035

Author(s):

Ilya Shmulevich

Keyword(s):

Lyapunov Exponent ◽

Dynamical Systems ◽

Boolean Functions ◽

Boolean Network ◽

Boolean Networks ◽

Asymptotic Formulas ◽

Small Perturbations ◽

Monotone Boolean Functions ◽

Almost All

Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.

Discrete Dynamical Systems: A Brief Survey

Journal of the Institute of Engineering ◽

10.3126/jie.v14i1.20067 ◽

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 ◽

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

Nanoscience and Nanotechnology Letters ◽

10.1166/nnl.2019.3042 ◽

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 ◽

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.

A new chaos synchronization criterion for discrete dynamical systems

Applied Mathematical Sciences ◽

10.12988/ams.2014.4132 ◽

2014 ◽

Vol 8 ◽

pp. 2025-2034 ◽

Cited By ~ 1

Author(s):

Adel Ouannas

Keyword(s):

Dynamical Systems ◽

Chaos Synchronization ◽

Discrete Dynamical Systems

Unfoldings of discrete dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700002558 ◽

1984 ◽

Vol 4 (3) ◽

pp. 421-486 ◽

Cited By ~ 1

Author(s):

Joel W. Robbin

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Conjugacy Class ◽

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.

STATISTICAL LAWS FOR COMPUTATIONAL COLLAPSE OF DISCRETIZED CHAOTIC MAPPINGS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127496001533 ◽

1996 ◽

Vol 06 (12a) ◽

pp. 2389-2399 ◽

Cited By ~ 8

Author(s):

PHIL DIAMOND ◽

ALEXEI POKROVSKII

Keyword(s):

Dynamical System ◽

Fixed Point ◽

Dynamical Systems ◽

Close Agreement ◽

Computer Experiments ◽

Continuum State ◽

Chaotic Dynamical Systems ◽

The Family ◽

Random Mappings ◽

Statistical Laws

When a dynamical system is realized on a computer, the computation is of a discretization, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretizations often have collapsing effects to a fixed point or to short cycles. Statistical properties of this phenomenon can be modeled by random mappings with an absorbing center. The model gives results which are very much in line with computational experiments and there appears to be a type of universality summarised by an Arcsine Law. The effects are discussed with special reference to the family of mappings fl(x)=1−|1−2x|l,x∈[0, 1], 1<l≤2. Computer experiments show close agreement with predictions of the model.

Bekenstein–Hawking entropy for discrete dynamical systems on sites

Modern Physics Letters A ◽

10.1142/s0217732319502651 ◽

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 ◽

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.