Dynamical Systems and their Chaotic Properties

2011 ◽  
pp. 1-18
Author(s):  
Kostas Chlouverakis
Keyword(s):  
Dynamical Systems ◽  
Chaotic System ◽  
Chaos Theory ◽  
Real World ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Chaotic Dynamical Systems ◽  
Real World Applications

This chapter describes the fundamental concepts of continuous-time chaotic dynamical systems so as to make the book self-contained and readable by fairly unfamiliar people with using simple chaotic systems and understanding their chaotic properties, but also by experts who desire to refine their knowledge regarding chaos theory. It will also define the sense in which a chaotic system is suitable for real-world applications such as encryption.

Random Bit Generator Based on Non-Autonomous Chaotic Systems

2015 ◽  
pp. 203-229
Author(s):  
Christos Volos ◽  
Ioannis Kyprianidis ◽  
Ioannis Stouboulos ◽  
Sundarapandian Vaidyanathan
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaos Theory ◽  
Chaotic Systems ◽  
Statistical Tests ◽  
Van Der Pol ◽  
Chaotic Dynamical Systems ◽  
Random Bit Generator ◽  
Close Relationship ◽  
Autonomous Dynamical System

In the last decade, a very interesting relationship between cryptography and chaos theory was developed. As a result of this close relationship, several chaos-based cryptosystems, especially using autonomous chaotic dynamical systems, have been put forward. However, this chapter presents a novel Chaotic Random Bit Generator (CRBG), which is based on the Poincaré map of a non-autonomous dynamical system. For this reason, the very-well known Duffing-van der Pol system has been used. The proposed CRBG also uses the X-OR function for improving the “randomness” of the produced bit streams, which are subjected to the most stringent statistical tests, the FIPS-140-2 suite tests, to detect the specific characteristics that are expected from random bit sequences.

scholarly journals A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization, and Its Digital Implementation

Inventions ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 49
Author(s):  
Zain-Aldeen S. A. Rahman ◽  
Basil H. Jasim ◽  
Yasir I. A. Al-Yasir ◽  
Raed A. Abd-Alhameed ◽  
Bilal Naji Alhasnawi
Keyword(s):  
Adaptive Control ◽  
Fractional Order ◽  
Chaotic System ◽  
Real World ◽  
Experimental Results ◽  
Chaotic Attractors ◽  
State Variables ◽  
Digital Implementation ◽  
Dynamical Behaviors ◽  
Real World Applications

In this paper, a new fractional order chaotic system without equilibrium is proposed, analytically and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigations were used to describe the system’s dynamical behaviors including the system equilibria, the chaotic attractors, the bifurcation diagrams, and the Lyapunov exponents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attractors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive control theory was developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state variables for the master and slave. Consequently, the update laws of the slave parameters are obtained, where the slave parameters are assumed to be uncertain and are estimated corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results were obtained by MATLAB and the Arduino Due boards, respectively, with a good consistency between the simulation results and the experimental results, indicating that the new fractional order chaotic system is capable of being employed in real-world applications.

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

scholarly journals A generic method to construct new customized-shaped haotic systems using the relative motion concept

2016 ◽  
Vol 21 (3) ◽  
pp. 413-423 ◽  
Author(s):  
Daniel-Ioan Curiac ◽  
Constantin Volosencu
Keyword(s):  
Chaotic System ◽  
Relative Motion ◽  
Real World ◽  
Chaotic Systems ◽  
Coordinate Frame ◽  
Closed Contour ◽  
Real World Application ◽  
Innovative Method ◽  
Stationary Frame

Constructing chaotic systems tailored for each particular real-world application has been a long-term research desideratum. We report a solution for this problem based on the concept of relative motion. We investigate the periodic motion on a closed contour of a coordinate frame in which a chaotic system evolves. By combining these two motions (periodic on a close contour and chaotic) new customized shape trajectories are acquired. We demonstrate that these trajectories obtained in the stationary frame are also chaotic and, moreover, conserve the Lyapunov exponents of the initial chaotic system. Based on this finding we developed an innovative method to construct new chaotic systems with customized shapes, thus fulfilling the requirements of any particular application of chaos.

scholarly journals A New Fractional-Order Chaotic System with Its Analysis, Synchronization, and Circuit Realization for Secure Communication Applications

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2593
Author(s):  
Zain-Aldeen S. A. Rahman ◽  
Basil H. Jasim ◽  
Yasir I. A. Al-Yasir ◽  
Yim-Fun Hu ◽  
Raed A. Abd-Alhameed ◽  
...  
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Real World ◽  
Communication System ◽  
Secure Communication ◽  
Communication Systems ◽  
Equilibrium Points ◽  
Lyapunov Theory ◽  
Uncertain Parameter ◽  
Real World Applications

This article presents a novel four-dimensional autonomous fractional-order chaotic system (FOCS) with multi-nonlinearity terms. Several dynamics, such as the chaotic attractors, equilibrium points, fractal dimension, Lyapunov exponent, and bifurcation diagrams of this new FOCS, are studied analytically and numerically. Adaptive control laws are derived based on Lyapunov theory to achieve chaos synchronization between two identical new FOCSs with an uncertain parameter. For these two identical FOCSs, one represents the master and the other is the slave. The uncertain parameter in the slave side was estimated corresponding to the equivalent master parameter. Next, this FOCS and its synchronization were realized by a feasible electronic circuit and tested using Multisim software. In addition, a microcontroller (Arduino Due) was used to implement the suggested system and the developed synchronization technique to demonstrate its digital applicability in real-world applications. Furthermore, based on the developed synchronization mechanism, a secure communication scheme was constructed. Finally, the security analysis metric tests were investigated through histograms and spectrograms analysis to confirm the security strength of the employed communication system. Numerical simulations demonstrate the validity and possibility of using this new FOCS in high-level security communication systems. Furthermore, the secure communication system is highly resistant to pirate attacks. A good agreement between simulation and experimental results is obtained, showing that the new FOCS can be used in real-world applications.

scholarly journals Fractional Variational Problems Depending on Fractional Derivatives of Differentiable Functions with Application to Nonlinear Chaotic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Matheus Jatkoske Lazo
Keyword(s):  
Dynamical Systems ◽  
Fractional Derivatives ◽  
Chaotic Systems ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Necessary Condition ◽  
Chaotic Dynamical Systems ◽  
Differentiable Functions ◽  
Fractional Variational Problems ◽  
Derivatives Of

We formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.

GEOMETRY OF TARGETING OF CHAOTIC SYSTEMS

1995 ◽  
Vol 05 (04) ◽  
pp. 1167-1173 ◽  
Author(s):  
M. PASKOTA ◽  
A. I. MEES ◽  
K. L. TEO
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Geometric Approach ◽  
Minimum Time ◽  
Chaotic Dynamical Systems

In this paper, we analyze the geometry of directing of orbits of chaotic dynamical systems. The geometric approach enables us to interpret the obtained results so as to complement some of the existing ideas about minimum-time targeting. The analysis is illustrated by an example.

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