Discrete Time Chaotic Maps With Application to Random Bit Generation

2021 ◽  
pp. 542-582
Author(s):  
Lazaros Moysis ◽  
Ahmad Taher Azar ◽  
Aleksandra Tutueva ◽  
Denis N. Butusov ◽  
Christos Volos
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Statistical Tests ◽  
Solution Space ◽  
Small Perturbations ◽  
Highly Sensitive ◽  
Different Shapes ◽  
Parameter Values

Chaotic behavior is a term that is attributed to dynamical systems whose solutions are highly sensitive to initial conditions. This means that small perturbations in the initial conditions can lead to completely different trajectories in the solution space. These types of chaotic dynamical systems arise in various natural or artificial systems in biology, circuits, engineering, computer science, and more. This chapter reports on some new chaotic discrete time two-dimensional maps that are derived from simple modifications to the well-known Hénon, Lozi, Sine-Sine, and Tinkerbell maps. Numerical simulations are carried out for different parameter values and initial conditions, and it is shown that the mappings either diverge to infinity or converge to attractors of many different shapes. The application to random bit generation is then considered using a collection of the proposed maps by applying a simple rule. The resulting bit generator successfully passes all statistical tests performed.

New Discrete Time 2D Chaotic Maps

2017 ◽  
Vol 6 (1) ◽  
pp. 77-104 ◽  
Author(s):  
Lazaros Moysis ◽  
Ahmad Taher Azar
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Single Point ◽  
Solution Space ◽  
Small Perturbations ◽  
Highly Sensitive ◽  
Different Shapes ◽  
Parameter Values

Chaotic behavior is a term that is attributed to dynamical systems whose solutions are highly sensitive to initial conditions. This means that small perturbations in the initial conditions can lead to completely different trajectories in the solution space. These types of chaotic dynamical systems arise in various natural or artificial systems in biology, meteorology, economics, electrical circuits, engineering, computer science and more. Of these innumerable chaotic systems, perhaps the most interesting are those that exhibit attracting behavior. By that, the authors refer to systems whose trajectories converge with time to a set of values, called an attractor. This can be a single point, a curve or a manifold. The attractor is called strange if it is a set with fractal structure. Such systems can be both continuous and discrete. This paper reports on some new chaotic discrete time two dimensional maps that are derived from simple modifications to the well-known Hénon, Lozi, Sine-sine and Tinkerbell maps. Numerical simulations are carried out for different parameter values and initial conditions and it is shown that the mappings either diverge to infinity or converge to attractors of many different shapes.

scholarly journals Stability, Bifurcation, Chaos : Discrete Prey Predator Model with Step Size

2019 ◽  
Vol 9 (1) ◽  
pp. 3382-3387 ◽  
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium States ◽  
Phase Portraits ◽  
Step Size ◽  
Dynamical Nature ◽  
Predator Model ◽  
The Stability ◽  
Parameter Values ◽  
Bifurcation Chaos

In this work titled Stability, Bifurcation, Chaos: Discrete prey predator model with step size, by Forward Euler Scheme method the discrete form is obtained. Equilibrium states are calculated and the stability of the equilibrium states and dynamical nature of the model are examined in the closed first quadrant 2 R with the help of variation matrix. It is observed that the system is sensitive to the initial conditions and also to parameter values. The dynamical nature of the model is investigated with the assistance of Lyapunov Exponent, bifurcation diagrams, phase portraits and chaotic behavior of the system is identified. Numerical simulations validate the theoretical observations.

CHUA’S EQUATION WITH CUBIC NONLINEARITY

1996 ◽  
Vol 06 (12a) ◽  
pp. 2175-2222 ◽  
Author(s):  
ANSHAN HUANG ◽  
LADISLAV PIVKA ◽  
CHAI WAH WU ◽  
MARTIN FRANZ
Keyword(s):  
Computer Program ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Tabular Form ◽  
Cubic Nonlinearity ◽  
Bifurcation Phenomena ◽  
Hands On ◽  
Autonomous Differential Equations ◽  
Parameter Values

In this tutorial paper we present one of the simplest autonomous differential equations capable of generating chaotic behavior. Some of the fundamental routes to chaos and bifurcation phenomena are demonstrated with examples. A brief discussion of equilibrium points and their stability is given. For the convenience of the reader, a short computer program written in QuickBASIC is included to give the reader a possibility of quick hands-on experience with the generation of chaotic phenomena without using sophisticated numerical simulators. All the necessary parameter values and initial conditions are provided in a tabular form. Eigenvalue diagrams showing regions with particular eigenvalue patterns are given.

STABILITY AND CHAOS IN REACTION SYSTEMS

2012 ◽  
Vol 23 (05) ◽  
pp. 1173-1184 ◽  
Author(s):  
ANDRZEJ EHRENFEUCHT ◽  
MICHAEL MAIN ◽  
GRZEGORZ ROZENBERG ◽  
ALLISON THOMPSON BROWN
Keyword(s):  
Experimental Study ◽  
Dynamical Systems ◽  
Living Cell ◽  
Chaotic Behavior ◽  
Discrete Dynamical Systems ◽  
Abstract Model ◽  
Biochemical Reactions ◽  
Initial State ◽  
Small Perturbations ◽  
Reaction Systems

Reaction systems are an abstract model of biochemical reactions in the living cell within a framework of finite (though often large) discrete dynamical systems. In this setting, this paper provides an analytical and experimental study of stability. The notion of stability is defined in terms of the way in which small perturbations to the initial state of a system are likely to change the system's eventual behavior. At the stable end of the spectrum, there is likely to be no change; but at the unstable end, small perturbations take the system into a state that is probabilistically the same as a randomly selected state, similar to chaotic behavior in continuous dynamical systems.

THE GENERALIZED HÉNON MAPS: EXAMPLES FOR HIGHER-DIMENSIONAL CHAOS

2002 ◽  
Vol 12 (06) ◽  
pp. 1371-1384 ◽  
Author(s):  
HENDRIK RICHTER
Keyword(s):  
Discrete Time ◽  
Finite Dimension ◽  
Initial Conditions ◽  
Henon Maps ◽  
Hénon Maps ◽  
Discrete Time Systems ◽  
Higher Dimensional ◽  
Parameter Values ◽  
Time Systems

The generalized Hénon maps (GHM) are discrete-time systems with given finite dimension, which show chaotic and hyperchaotic behavior for certain parameter values and initial conditions. A study of these maps is given where particularly higher-dimensional cases are considered.

scholarly journals ESTUDO DO COMPORTAMENTO DINÂMICO DO MODELO NEURONAL DE HINDMARSH-ROSE

2019 ◽  
Vol 11 (4) ◽  
pp. 122-130
Author(s):  
RaildoSantos de Lima ◽  
Fábio Roberto Chavarette ◽  
Luiz Gustavo Pereira Roéfero Roéfero
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Nerve Impulse ◽  
Neuronal Model ◽  
Random Behavior ◽  
Highly Sensitive ◽  
Real Neuron ◽  
Hr Model ◽  
Biological Neuron ◽  
The Stability

Based on the Hindmarsh-Rose (RH) neuronal model for nerve impulse transmission, this paper aims to study the properties and dynamic behavior of the non-linear chaotic system that describes neuronal bursting in a single neuron. On the part of bioengineering, there is great motivation in the study of the HR model because it is well representative of the biological neuron, being able to simulate several behaviors of a real neuron, among them periodic, aperiodic and chaotic behavior. The literature suggests that the chaotic behaviorrepresents in the human being the epileptic or convulsive state. Through computer simulations, considering the system parameters, it was analyzed that the stability is highly sensitive to the initial conditions and producing oscillations, more so, when the oscillation increases the random behavior tends to increase making the system unpredictable.

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.

Chaotic behavior of discrete-time linear inclusion dynamical systems

2017 ◽  
Vol 354 (10) ◽  
pp. 4126-4155
Author(s):  
Xiongping Dai ◽  
Tingwen Huang ◽  
Yu Huang ◽  
Yi Luo ◽  
Gang Wang ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Linear Inclusion ◽  
Time Linear

scholarly journals Stability Analysis of the Chaotic Lorenz System with a State-Feedback Controller

2021 ◽  
Author(s):  
Dan Jones
Keyword(s):  
Strange Attractor ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Geometric Approach ◽  
Stable Equilibrium Point ◽  
The Lorenz System ◽  
The Stability ◽  
Parameter Values

The Lorenz model is considered a benchmark system in chaotic dynamics in that it displays extraordinary sensitivity to initial conditions and the strange attractor phenomenon. Even though the system tends to amplify perturbations, it is indeed possible to convert a strange attractor to a non-chaotic one using various control schemes. In this work it is shown that the chaotic behavior of the Lorenz system can be suppressed through the use of a feedback loop driven by a quotient controller. The stability of the controlled Lorenz system is evaluated near its equilibrium points using Routh-Hurwitz testing, and the global stability of the controlled system is established using a geometric approach. It is shown that the controlled Lorenz system has only one globally stable equilibrium point for the set of parameter values under consideration.

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