The Features of Chaos

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Average Rate ◽  
Quantitative Measure ◽  
Fractal Dimensions ◽  
Recurrence Plot ◽  
Sensitivity To Initial Conditions

In this chapter we introduce the features of Chaotic systems. We describe “sensitivity to initial conditions” and its quantitative measure, the Lyapunov exponent, which reflect the average rate of divergence (if any) between two neighboring trajectories. We describe the dynamic “strangeness” of the system. Which has its counterpart in the “strangeness” of the attractor's geometry and concerns with the texture woven by the system in phase space. Fractal dimensions are measures of such strange geometries and they are here described. The concept of recurrence is introduced and the recurrence plot is described, and code provided to generate it. The correlation dimension is addressed and the R code to compute is listed and detailed. Poincare map is introduced and applied to the study of the damped, driven pendulum.

scholarly journals Experimental Realization of a Multiscroll Chaotic Oscillator with Optimal Maximum Lyapunov Exponent

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Esteban Tlelo-Cuautle ◽  
Ana Dalia Pano-Azucena ◽  
Victor Hugo Carbajal-Gomez ◽  
Mauro Sanchez-Sanchez
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Mathematical Description ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Quantitative Measure ◽  
Operational Amplifiers ◽  
Chaotic Oscillator ◽  
Maximum Lyapunov Exponent ◽  
Experimental Realization

Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. However, those realizations do not consider the value of the maximum Lyapunov exponent, which gives a quantitative measure of the grade of unpredictability of chaotic systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. First, from the mathematical description having four coefficients (a, b, c, d1), an optimization evolutionary algorithm varies them to maximize the value of the positive Lyapunov exponent. Second, a realization of those optimized coefficients using operational amplifiers is given. Hereina, b, c, d1are implemented with precision potentiometers to tune up to four decimals of the coefficients having the range between 0.0001 and 1.0000. Finally, experimental results of the phase-space portraits for generating from 2 to 10 scrolls are listed to show that their associated value for the optimal maximum Lyapunov exponent increases by increasing the number of scrolls, thus guaranteeing a more complex chaotic behavior.

A New Image Encryption Scheme Using Dual Chaotic Map Synchronization

2020 ◽  
Vol 18 (1) ◽  
Keyword(s):  
Image Encryption ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Chaotic Map ◽  
Security Applications ◽  
The Common ◽  
And Diffusion ◽  
Confusion And Diffusion ◽  
Sensitivity To Initial Conditions ◽  
Diffusion Properties

Chaotic systems behavior attracts many researchers in the field of image encryption. The major advantage of using chaos as the basis for developing a crypto-system is due to its sensitivity to initial conditions and parameter tunning as well as the random-like behavior which resembles the main ingredients of a good cipher namely the confusion and diffusion properties. In this article, we present a new scheme based on the synchronization of dual chaotic systems namely Lorenz and Chen chaotic systems and prove that those chaotic maps can be completely synchronized with other under suitable conditions and specific parameters that make a new addition to the chaotic based encryption systems. This addition provides a master-slave configuration that is utilized to construct the proposed dual synchronized chaos-based cipher scheme. The common security analyses are performed to validate the effectiveness of the proposed scheme. Based on all experiments and analyses, we can conclude that this scheme is secure, efficient, robust, reliable, and can be directly applied successfully for many practical security applications in insecure network channels such as the Internet

ENERGY-LIKE FUNCTIONS FOR SOME DISSIPATIVE CHAOTIC SYSTEMS

2005 ◽  
Vol 15 (08) ◽  
pp. 2507-2521 ◽  
Author(s):  
C. SARASOLA ◽  
A. D'ANJOU ◽  
F. J. TORREALDEA ◽  
A. MOUJAHID
Keyword(s):  
Phase Space ◽  
Dynamic Behavior ◽  
Fourth Order ◽  
Chaotic Systems ◽  
Quantitative Measure ◽  
Energy Functions ◽  
Dissipation Of Energy ◽  
Rossler System ◽  
Order Polynomial ◽  
Fourth Order Polynomial

Functions of the phase space variables that can considered as possible energy functions for a given family of dissipative chaotic systems are discussed. This kind of functions are interesting due to their use as an energy-like quantitative measure to characterize different aspects of dynamic behavior of associated chaotic systems. We have calculated quadratic energy-like functions for the cases of Lorenz, Chen, Lü–Chen and Chua, and show the patterns of dissipation of energy on their respective attractors. We also show that in the case of the Rössler system at least a fourth-order polynomial is required to properly represent its energy.

Sensitivity to initial conditions and lyapunov exponent of a quasiperiodic system

2000 ◽  
Vol 45 (5) ◽  
pp. 633-635
Author(s):  
I. A. Khovanov ◽  
N. A. Khovanova ◽  
V. S. Anishchenko ◽  
P. W. E. McClintock
Keyword(s):  
Lyapunov Exponent ◽  
Initial Conditions ◽  
Sensitivity To Initial Conditions

scholarly journals Sources of uncertainty in deterministic dynamics: an informal overview

2011 ◽  
Vol 369 (1956) ◽  
pp. 4705-4729 ◽  
Author(s):  
Ian Stewart
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Sources Of Uncertainty ◽  
Deterministic Systems ◽  
Coin Tossing ◽  
Random Switching ◽  
Butterfly Effect ◽  
Basin Boundaries ◽  
Sensitivity To Initial Conditions

The discovery of chaotic dynamics implies that deterministic systems may not be predictable in any meaningful sense. The best-known source of unpredictability is sensitivity to initial conditions (popularly known as the butterfly effect), in which small errors or disturbances grow exponentially. However, there are many other sources of uncertainty in nonlinear dynamics. We provide an informal overview of some of these, with an emphasis on the underlying geometry in phase space. The main topics are the butterfly effect, uncertainty in initial conditions in non-chaotic systems, such as coin tossing, heteroclinic connections leading to apparently random switching between states, topological complexity of basin boundaries, bifurcations (popularly known as tipping points) and collisions of chaotic attractors. We briefly discuss possible ways to detect, exploit or mitigate these effects. The paper is intended for non-specialists.

LOW-DIMENSIONAL BEHAVIOR OF NONLINEAR TEARING MODE DYNAMICS IN A ROTATING PLASMA

1993 ◽  
Vol 03 (05) ◽  
pp. 1155-1168 ◽  
Author(s):  
M. PERSSON ◽  
C. R. LAING
Keyword(s):  
Lyapunov Exponent ◽  
Initial Conditions ◽  
Largest Lyapunov Exponent ◽  
Tearing Mode ◽  
Oscillatory State ◽  
The Largest Lyapunov Exponent ◽  
Low Dimensional ◽  
Sensitivity To Initial Conditions

Simulations of resistive magnetohydrodynamics in a rotating plasma are analyzed by calculating the correlation dimensions and the local intrinsic dimensions. A rotating plasma with a nonlinear oscillatory state is found to be associated with a low-dimensional attractor. The solutions are also checked for sensitivity to initial conditions, characterized by the sign of the largest Lyapunov exponent.

On the Correlation Dimension of Discrete Fractional Chaotic Systems

2020 ◽  
Vol 30 (12) ◽  
pp. 2050174 ◽  
Author(s):  
Li Ma ◽  
Xianggang Liu ◽  
Xiaotong Liu ◽  
Ying Zhang ◽  
Yu Qiu ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Chaotic Systems ◽  
Jacobian Matrix ◽  
Three Dimensional ◽  
Largest Lyapunov Exponent ◽  
Embedding Dimension ◽  
Chaotic State ◽  
Fractional Systems ◽  
With Memory

This paper is mainly devoted to the investigation of discrete-time fractional systems in three aspects. Firstly, the fractional Bogdanov map with memory effect in Riemann–Liouville sense is obtained. Then, via constructing suitable controllers, the fractional Bogdanov map is shown to undergo a transition from regular state to chaotic one. Meanwhile, the positive largest Lyapunov exponent is calculated by the Jacobian matrix algorithm to distinguish the chaotic areas. Finally, the Grassberger–Procaccia algorithm is employed to evaluate the correlation dimension of the controlled fractional Bogdanov system under different parameters. The main results show that the correlation dimension converges to a fixed value as the embedding dimension increases for the controlled fractional Bogdanov map in chaotic state, which also coincides with the conclusion driven by the largest Lyapunov exponent. Moreover, three-dimensional fractional Stefanski map is considered to further verify the effectiveness and generality of the obtained results.

Control Lorenz System with Vibration Estimation with Averaging Method

2010 ◽  
Vol 43 ◽  
pp. 36-39
Author(s):  
Chun Zhou
Keyword(s):  
Chaotic Systems ◽  
Inverted Pendulum ◽  
Lorenz System ◽  
Stable Equilibrium ◽  
Averaging Method ◽  
Control Method ◽  
Initial Conditions ◽  
Vibrational Control ◽  
Extreme Sensitivity ◽  
Sensitivity To Initial Conditions

The vibrational control theory stems from the well-known of stabilization of the upper unstable equilibrium position of the inverted pendulum having suspension point vibration along the vertical line with amplitude as small as desired and a frequency reason high. Chaotic phenomena have been found in many nonlinear systems including continuous time and discrete time. The chaotic systems are characterized by their extreme sensitivity to initial conditions, nonperiodic and boundary. The trajectories start even from close initial states will diverge from each other at an exponential rate as time goes. The vibrational control method was applied to Lorenz system. The effect of the control can be estimated with the APAZ method. It was showed that vibrational control brought the controlled Lorenz system to stable equilibrium with appropriate parameters. Numerical simulation demonstrated validity of the proposed method.

The Chaotic Attractor of the Sediment Movement

Fractals ◽  
1998 ◽  
Vol 06 (02) ◽  
pp. 191-196
Author(s):  
Fengsu Chen ◽  
Kongxian Xue ◽  
Wenkang Cai
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Chaotic Attractor ◽  
Yangtze River ◽  
Correlation Dimension ◽  
Chaotic Behavior ◽  
Largest Lyapunov Exponent ◽  
The Yangtze River ◽  
Sediment Movement ◽  
Reconstructed Phase Space

We consider the chaotic behavior of the sediment movement with the observed data of the Yangtze River in China and the method of the reconstructed phase space and we find that in the sediment movement there is an attractor. As far as the real example mentioned in this paper is concerned, the correlation dimension and the largest Lyapunov exponent are around 6.6 and 0.013 respectively. These results are crucially referential for estimating the mode of the sediment movement, designing the scheme of the sediment observation, and studying the predictability problem of the sediment.

scholarly journals A history of chaos theory

2007 ◽  
Vol 9 (3) ◽  
pp. 279-289 ◽  
Keyword(s):  
Mathematical Models ◽  
Chaos Theory ◽  
Cardiac Arrhythmias ◽  
Initial Conditions ◽  
Fractal Dimensions ◽  
Biological Clocks ◽  
Deterministic Models ◽  
Endogenous Rhythms ◽  
History Of ◽  
Sensitivity To Initial Conditions

Whether every effect can be precisely linked to a given cause or to a list of causes has been a matter of debate for centuries, particularly during the 17th century, when astronomers became capable of predicting the trajectories of planets. Recent mathematical models applied to physics have included the idea that given phenomena cannot be predicted precisely, although they can be predicted to some extent, in line with the chaos theory. Concepts such as deterministic models, sensitivity to initial conditions, strange attractors, and fractal dimensions are inherent to the development of this theory A few situations involving normal or abnormal endogenous rhythms in biology have been analyzed following the principles of chaos theory. This is particularly the case with cardiac arrhythmias, but less so with biological clocks and circadian rhythms.

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