Modeling and Imaging Mechanical Chaos

2005 ◽  
Author(s):  
Mike Bailley
Keyword(s):  
Mechanical System ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Complex Nature ◽  
Dynamics Modeling ◽  
System A

The word “chaotic system” [Peitgen92] describes a system whose outputs are very sensitive to its initial conditions. Because of their inherent complex nature, chaotic systems are difficult to visualize and understand. This paper describes the visualization of a mechanical chaotic system — a magnetic pendulum. The program uses dynamics modeling and imaging, so that a user can experiment with different configurations and then visualize how that configuration responds to all input conditions. The result shows interesting patterns and insights into the mechanical system itself. This same technique would be applicable to visualizing many other chaotic systems.

scholarly journals A New 3D Autonomous Continuous System with Two Isolated Chaotic Attractors and Its Topological Horseshoes

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Ping Zhou ◽  
Meihua Ke
Keyword(s):  
Numerical Computation ◽  
Topological Entropy ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Continuous System ◽  
Chaotic Attractors ◽  
System A ◽  
Topological Horseshoes

Based on the 3D autonomous continuous Lü chaotic system, a new 3D autonomous continuous chaotic system is proposed in this paper, and there are coexisting chaotic attractors in the 3D autonomous continuous chaotic system. Moreover, there are no overlaps between the coexisting chaotic attractors; that is, there are two isolated chaotic attractors (in this paper, named “positive attractor” and “negative attractor,” resp.). The “positive attractor” and “negative attractor” depend on the distance between the initial points (initial conditions) and the unstable equilibrium points. Furthermore, by means of topological horseshoes theory and numerical computation, the topological horseshoes in this 3D autonomous continuous system is found, and the topological entropy is obtained. These results indicate that the chaotic attractor emerges in the new 3D autonomous continuous system.

Hidden Attractor and Multistability in a Novel Memristor-Based System Without Symmetry

2021 ◽  
Vol 31 (11) ◽  
pp. 2150168
Author(s):  
Musha Ji’e ◽  
Dengwei Yan ◽  
Lidan Wang ◽  
Shukai Duan
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Control Unit ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Dynamic Behaviors ◽  
Potential Applications

Memristor, as a typical nonlinear element, is able to produce chaotic signals in chaotic systems easily. Chaotic systems have potential applications in secure communications, information encryption, and other fields. Therefore, it is of importance to generate abundant dynamic behaviors in a single chaotic system. In this paper, a novel memristor-based chaotic system without equilibrium points is proposed. One of the essential features is the absence of symmetry in this system, which increases the complexity of the new system. Then, the nonlinear dynamic behaviors of the system are analyzed in terms of chaos diagrams, bifurcation diagrams, Poincaré maps, Lyapunov exponent spectra, the sum of Lyapunov exponents, phase portraits, 0–1 test, recurrence analysis and instantaneous phase. The results of the sum of Lyapunov exponents show that the given system is a quasi-Hamiltonian system with certain initial conditions (IC) and parameters. Next, other critical phenomena, such as hidden multi-scroll attractors, abundant coexistence characteristics, are found characterized through basins of attraction and others. Especially, it reveals some rare phenomena in other systems that multiple hidden hyperchaotic attractors coexist. Finally, the circuit implementation based on Micro Control Unit (MCU) confirms theoretical analysis and the numerical simulation.

Simultaneous Time-Frequency Synchronization of Non-Autonomous Chaotic Systems

2013 ◽  
Author(s):  
Meng-Kun Liu ◽  
C. Steve Suh
Keyword(s):  
Chaotic System ◽  
Initial Conditions ◽  
Frequency Control ◽  
Frequency Synchronization ◽  
Time Frequency ◽  
System A ◽  
Frequency Domains ◽  
Control Concept ◽  
On Line ◽  
The Time Domain

A novel chaos control concept is presented for the synchronization of a non-autonomous chaotic circuit system in the time and frequency domains concurrently. The controller effectively eliminates the differences between two chaotic circuits in the time domain and at the same time restores the characteristics of the driving response in the frequency domain. The simultaneous time-frequency control is achieved through manipulating wavelet coefficients, thus not limited by the increasing bandwidth of the chaotic system — a fundamental restraint that deprives contemporary controller designs of validity and effectiveness. The feedforward feature of the control concept prevents errors from re-entering the control loop and inadvertently perturbing the sensitive chaotic system. Because neither closed-form nor linearization is required, the innate, genuine features of the chaotic response are faithfully retained. The on-line identification feature allows the response system to start at arbitrary initial conditions and to be driven by the sinusoidal forcing term of different amplitudes and phases requiring no knowledge of the system parameters.

One-to-four-wing hyperchaotic fractional-order system and its circuit realization

Circuit World ◽  
2020 ◽  
Vol 46 (2) ◽  
pp. 107-115
Author(s):  
Xiang Li ◽  
Zhijun Li ◽  
Zihao Wen
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Complexity Analysis ◽  
Initial Conditions ◽  
Fractional Order System ◽  
Order System ◽  
Hyperchaotic System ◽  
Circuit Realization ◽  
Content Type

Purpose This paper aims to introduce a novel 4D hyperchaotic fractional-order system which can produce one-to-four-wing hyperchaotic attractors. In the study of chaotic systems with variable-wing attractors, although some chaotic systems can generate one-to-four-wing attractors, none of them are hyperchaotic attractors, which is incomplete for the dynamic characteristics of chaotic systems. Design/methodology/approach A novel 4D fractional-order hyperchaotic system is proposed based on the classical three-dimensional Lü system. The complex and abundant dynamic behaviors of the fractional-order system are analyzed by phase diagrams, bifurcation diagrams and the corresponding Lyapunov exponents. In addition, SE and C0 algorithms are used to analyze the complexity of the fractional-order system. Then, the influence of order q on the system is also investigated. Finally, the circuit is implemented using physical components. Findings The most particular interest is that the system can generate one-to-four-wing hyperchaotic attractors with only one parameter variation. Then, the hardware circuit experimental results tally with the numerical simulations, which proves the validity and feasibility of the fractional-order hyperchaotic system. Besides, under different initial conditions, coexisting attractors can be obtained by changing the parameter d or the order q. Then, the complexity analysis of the system shows that the fractional-order chaotic system has higher complexity than the corresponding integer-order chaotic system. Originality/value The circuit structure of the fractional-order hyperchaotic system is simple and easy to implement, and one-to-four-wing hyperchaotic attractors can be observed in the circuit. To the best of the knowledge, this unique phenomenon has not been reported in any literature. It is of great reference value to analysis and circuit realization of fractional-order chaotic systems.

scholarly journals Anti-Synchronization of a Class of Chaotic Systems with Application to Lorenz System: A Unified Analysis of the Integer Order and Fractional Order

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 559 ◽  
Author(s):  
Liang Chen ◽  
Chengdai Huang ◽  
Haidong Liu ◽  
Yonghui Xia
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Finite Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Unknown Parameters ◽  
Integer Order ◽  
System A ◽  
Control Scheme ◽  
Lorenz Chaotic System

The paper proves a unified analysis for finite-time anti-synchronization of a class of integer-order and fractional-order chaotic systems. We establish an effective controller to ensure that the chaotic system with unknown parameters achieves anti-synchronization in finite time under our controller. Then, we apply our results to the integer-order and fractional-order Lorenz system, respectively. Finally, numerical simulations are presented to show the feasibility of the proposed control scheme. At the same time, through the numerical simulation results, it is show that for the Lorenz chaotic system, when the order is greater, the more quickly is anti-synchronization achieved.

Attack on a Chaos-Based Random Number Generator Using Anticipating Synchronization

2015 ◽  
Vol 25 (02) ◽  
pp. 1550021
Author(s):  
Ramazan Yeniçeri ◽  
Selçuk Kilinç ◽  
Müştak E. Yalçin
Keyword(s):  
Time Delay ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Random Number ◽  
Initial Conditions ◽  
Random Number Generator ◽  
Original System ◽  
Coupled Systems ◽  
Number Generator ◽  
Delay Chaotic System

Chaotic systems have been used in random number generation, owing to the property of sensitive dependence on initial conditions and hence the possibility to produce unpredictable signals. Within the types of chaotic systems, those which are defined by only one delay-differential equation are attractive due to their simple model. On the other hand, it is possible to synchronize to the future states of a time-delay chaotic system by anticipating synchronization. Therefore, random number generator (RNG), which employs such a system, might not be immune to the attacks. In this paper, attack on a chaos-based random number generator using anticipating synchronization is investigated. The considered time-delay chaotic system produces binary signals, which can directly be used as a source of RNG. Anticipating synchronization is obtained by incorporating other systems appropriately coupled to the original one. Quantification of synchronization is given by the bit error between the streams produced by the original and coupled systems. It is shown that the bit streams generated by the original system can be anticipated by the coupled systems beforehand.

scholarly journals Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 556 ◽  
Author(s):  
Shaobo He ◽  
Chunbiao Li ◽  
Kehui Sun ◽  
Sajad Jafari
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Complexity Analysis ◽  
Initial Conditions ◽  
Scale Factor ◽  
The Self ◽  
Permutation Entropy ◽  
Chaotic Attractors

Designing a chaotic system with infinitely many attractors is a hot topic. In this paper, multiscale multivariate permutation entropy (MMPE) and multiscale multivariate Lempel–Ziv complexity (MMLZC) are employed to analyze the complexity of those self-reproducing chaotic systems with one-directional and two-directional infinitely many chaotic attractors. The analysis results show that complexity of this class of chaotic systems is determined by the initial conditions. Meanwhile, the values of MMPE are independent of the scale factor, which is different from the algorithm of MMLZC. The analysis proposed here is helpful as a reference for the application of the self-reproducing systems.

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

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

Soft Computing Simulations of Chaotic Systems

2019 ◽  
Vol 29 (08) ◽  
pp. 1950112 ◽  
Author(s):  
Erivelton G. Nepomuceno ◽  
Priscila F. S. Guedes ◽  
Alípio M. Barbosa ◽  
Matjaž Perc ◽  
Robert Repnik
Keyword(s):  
Lower Bound ◽  
Soft Computing ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Logistic Map ◽  
Largest Lyapunov Exponent ◽  
Lower And Upper Bounds ◽  
Finite Precision ◽  
Chua Circuit ◽  
Widespread Interest

Soft computing strategies are drawing widespread interest in engineering and science fields, particularly so because of their capacity to reason and learn in a domain of inherent uncertainty, approximation, and unpredictability. However, soft computing research devoted to finite precision effects in chaotic system simulations is still in a nascent stage, and there are ample opportunities for new discoveries. In this paper, we consider the error that is due to finite precision in the simulation of chaotic systems. We present a generalized version of the lower bound error using an arbitrary number of natural interval extensions. The lower bound error has been used to simulate a chaotic system with lower and upper bounds. The width of this interval does not diverge, which is an advantage compared to other techniques. We illustrate our approach on three systems, namely the logistic map, the Singer map and the Chua circuit. Moreover, we validate the method by calculating the largest Lyapunov exponent.

Sign in / Sign up

Export Citation Format

Share Document