COMPOUND CHAOS

1996 ◽  
Vol 06 (02) ◽  
pp. 383-393 ◽  
Author(s):  
R. SINGH ◽  
P. S. MOHARIR ◽  
V. M. MARU
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Limit Cycles ◽  
Periodic System ◽  
Chaotic Systems ◽  
Deterministic Chaos ◽  
Random Variables ◽  
Statistical Distribution ◽  
Quantitative Studies ◽  
Compound Distribution

Compounding is a statistical notion. Essentially, it comprises of regarding the parameters in a particular statistical distribution as random variables with a prescribed distribution. The compound distribution then acquires the parameters of the compounding distribution as its own. As deterministic chaos, in spite of being deterministic, appears like a statistical phenomenon, the notion of compounding can be extended to chaotic systems. It is shown with illustrations that a chaotic system can be compounded by another chaotic system, giving rise to compound chaos which is, in general, “chaoticer”. The concept can also be used to make a periodic system chaotic, thus opening possibilities of “chaoticization”. Examples of compound chaos and chaoticization are given using Lorenz and Rössler systems, including their attractors and limit cycles as “compoundee” and/or “compounder” systems. The conclusions are based on quantitative studies of Lyapunov exponents and correlation dimensions.

MUTUAL AND SELF-COMPOUNDING OF CHAOTIC SYSTEMS

1996 ◽  
Vol 06 (04) ◽  
pp. 759-767
Author(s):  
R. SINGH ◽  
P.S. MOHARIR ◽  
V.M. MARU
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Dynamic Systems ◽  
Mantle Convection ◽  
Chaotic Systems ◽  
Compound System ◽  
Two Systems

The notion of compounding a chaotic system was introduced earlier. It consisted of varying the parameters of the compoundee system in proportion to the variables of the compounder system, resulting in a compound system which has in general higher Lyapunov exponents. Here, the notion is extended to self-compounding of a system with a real-earth example, and mutual compounding of dynamic systems. In the former, the variables in a system perturb its parameters. In the latter, two systems affect the parameters of each other in proportion to their variables. Examples of systems in such compounding relationships are studied. The existence of self-compounding is indicated in the geodynamics of mantle convection. The effect of mutual compounding is studied in terms of Lyapunov exponent variations.

scholarly journals Hybrid Passivity Based and Fuzzy Type-2 Controller for Chaotic and Hyper-Chaotic Systems

2017 ◽  
Vol 11 (2) ◽  
pp. 96-103 ◽  
Author(s):  
Fernando Serrano ◽  
Josep M. Rossell
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Control Strategy ◽  
Chaotic Systems ◽  
Control Law ◽  
Four Dimensions ◽  
Design Requirements ◽  
The Stability ◽  
Theoretical Results

AbstractIn this paper a hybrid passivity based and fuzzy type-2 controller for chaotic and hyper-chaotic systems is presented. The proposed control strategy is an appropriate choice to be implemented for the stabilization of chaotic and hyper-chaotic systems due to the energy considerations of the passivity based controller and the flexibility and capability of the fuzzy type-2 controller to deal with uncertainties. As it is known, chaotic systems are those kinds of systems in which one of their Lyapunov exponents is real positive, and hyper-chaotic systems are those kinds of systems in which more than one Lyapunov exponents are real positive. In this article one chaotic Lorentz attractor and one four dimensions hyper-chaotic system are considered to be stabilized with the proposed control strategy. It is proved that both systems are stabilized by the passivity based and fuzzy type-2 controller, in which a control law is designed according to the energy considerations selecting an appropriate storage function to meet the passivity conditions. The fuzzy type-2 controller part is designed in order to behave as a state feedback controller, exploiting the flexibility and the capability to deal with uncertainties. This work begins with the stability analysis of the chaotic Lorentz attractor and a four dimensions hyper-chaotic system. The rest of the paper deals with the design of the proposed control strategy for both systems in order to design an appropriate controller that meets the design requirements. Finally, numerical simulations are done to corroborate the obtained theoretical results.

Exploiting the controlled responses of chaotic elements to design configurable hardware

2006 ◽  
Vol 364 (1846) ◽  
pp. 2483-2494 ◽  
Author(s):  
Sudeshna Sinha ◽  
William L Ditto
Keyword(s):  
Fixed Points ◽  
Chaotic System ◽  
Limit Cycles ◽  
Chaotic Systems ◽  
Dynamic Logic ◽  
Logic Gates ◽  
Threshold Level ◽  
Experimental Realization ◽  
Configurable Hardware ◽  
Simple Manipulation

We discuss how threshold mechanisms can be effectively employed to control chaotic systems onto stable fixed points and limit cycles of widely varying periodicities. Then, we outline the theory and experimental realization of fundamental logic-gates from a chaotic system, using thresholding to effect control. A key feature of this implementation is that a single chaotic ‘processor’ can be flexibly configured (and re-configured) to emulate different fixed or dynamic logic gates through the simple manipulation of a threshold level.

SYNCHRONIZATION THEOREM FOR A CHAOTIC SYSTEM

1995 ◽  
Vol 05 (01) ◽  
pp. 297-302 ◽  
Author(s):  
JÖRG SCHWEIZER ◽  
MICHAEL PETER KENNEDY ◽  
MARTIN HASLER ◽  
HERVÉ DEDIEU
Keyword(s):  
Lyapunov Function ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Secure Communication ◽  
Communication Systems ◽  
Chaotic Systems ◽  
Secure Communications ◽  
Error Feedback ◽  
Response System ◽  
Synchronization Method

Since Pecora & Carroll [Pecora & Carroll, 1991; Carroll & Pecora, 1991] have shown that it is possible to synchronize chaotic systems by means of a drive-response partition of the systems, various authors have proposed synchronization schemes and possible secure communications applications [Dedieu et al., 1993, Oppenheim et al., 1992]. In most cases synchronization is proven by numerically computing the conditional Lyapunov exponents of the response system. In this work a new synchronization method using error-feedback is developed, where synchronization is provable using a global Lyapunov function. Furthermore, it is shown how this scheme can be applied to secure communication systems.

A GENERALIZED 3-D FOUR-WING CHAOTIC SYSTEM

2009 ◽  
Vol 19 (11) ◽  
pp. 3841-3853 ◽  
Author(s):  
ZENGHUI WANG ◽  
GUOYUAN QI ◽  
YANXIA SUN ◽  
MICHAËL ANTONIE VAN WYK ◽  
BAREND JACOBUS VAN WYK
Keyword(s):  
Phase Diagrams ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Autonomous System ◽  
Chaotic Systems ◽  
Three Dimensional ◽  
Bifurcation Diagrams ◽  
Basic Properties ◽  
New System

In this paper, several three-dimensional (3-D) four-wing smooth quadratic autonomous chaotic systems are analyzed. It is shown that these systems have similar features. A simpler and generalized 3-D continuous autonomous system is proposed based on these features which can be extended to existing 3-D four-wing chaotic systems by adding some linear and/or quadratic terms. The new system can generate a four-wing chaotic attractor with simple topological structures. Some basic properties of the new system is analyzed by means of Lyapunov exponents, bifurcation diagrams and Poincaré maps. Phase diagrams show that the equilibria are related to the existence of multiple wings.

scholarly journals A Torus-Chaotic System and Its Pseudorandom Properties

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Jizhao Liu ◽  
Xiangzi Zhang ◽  
Qingchun Zhao ◽  
Jing Lian ◽  
Fangjun Huang ◽  
...  
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Cyber Security ◽  
Secure Communication ◽  
Chaotic Systems ◽  
Random Number ◽  
Current Knowledge ◽  
Random Number Generator ◽  
Qr Code ◽  
Security Systems

Exploring and investigating new chaotic systems is a popular topic in nonlinear science. Although numerous chaotic systems have been introduced in the literature, few of them focus on torus-chaotic system. The aim of our short work is to widen the current knowledge of torus chaos. In this paper, a new torus-chaotic system is proposed, which has one positive Lyapunov exponent, two zero Lyapunov exponents, and two negative Lyapunov exponents. The dynamic behavior is investigated by Lyapunov exponents, bifurcations, and stability. The analysis shows that this system has an interesting route leading to chaos. Furthermore, the pseudorandom properties of output sequence are well studied and a random number generator algorithm is proposed, which has the potential of being used in several cyber security systems such as the verification code, secure QR code, and some secure communication protocols.

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.

Constructing a Novel No-Equilibrium Chaotic System

2014 ◽  
Vol 24 (05) ◽  
pp. 1450073 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Zhouchao Wei ◽  
Xiong Wang
Keyword(s):  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Poincaré Map ◽  
Period Doubling ◽  
Circuit Realization ◽  
Chaotic Flow ◽  
Route To Chaos ◽  
Line Equilibrium

This paper introduces a new no-equilibrium chaotic system that is constructed by adding a tiny perturbation to a simple chaotic flow having a line equilibrium. The dynamics of the proposed system are investigated through Lyapunov exponents, bifurcation diagram, Poincaré map and period-doubling route to chaos. A circuit realization is also represented. Moreover, two other new chaotic systems without equilibria are also proposed by applying the presented methodology.

RELATIVE KOLMOGOROV ENTROPY OF A CHAOTIC SYSTEM IN THE PRESENCE OF NOISE

2008 ◽  
Vol 18 (09) ◽  
pp. 2851-2855 ◽  
Author(s):  
VADIM S. ANISHCHENKO ◽  
SERGEY ASTAKHOV
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Stochastic System ◽  
Chaotic Systems ◽  
Metric Entropy ◽  
Kolmogorov Entropy ◽  
Entropy Estimation ◽  
Relative Metric ◽  
Mixing Property

The mixing property is characterized by the metric entropy that is introduced by Kolmogorov for dynamical systems. The Kolmogorov entropy is infinite for a stochastic system. In this work, a relative metric entropy is considered. The relative metric entropy allows to estimate the level of mixing in noisy dynamical systems. An algorithm for calculating the relative metric entropy is described and examples of the metric entropy estimation are provided for certain chaotic systems with various noise intensities. The results are compared to the entropy estimation given by the positive Lyapunov exponents.

scholarly journals Constructing Discrete Chaotic Systems with Positive Lyapunov Exponents

2018 ◽  
Vol 28 (07) ◽  
pp. 1850084 ◽  
Author(s):  
Chuanfu Wang ◽  
Chunlei Fan ◽  
Qun Ding
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Secure Communication ◽  
Discrete System ◽  
Chaotic Systems ◽  
Hyperchaotic System ◽  
Universal Method ◽  
Periodic Attractors ◽  
Chaotic Secure Communication ◽  
Hyperchaotic Systems

The chaotic system is widely used in chaotic cryptosystem and chaotic secure communication. In this paper, a universal method for designing the discrete chaotic system with any desired number of positive Lyapunov exponents is proposed to meet the needs of hyperchaotic systems in chaotic cryptosystem and chaotic secure communication, and three examples of eight-dimensional discrete system with chaotic attractors, eight-dimensional discrete system with fixed point attractors and eight-dimensional discrete system with periodic attractors are given to illustrate how the proposed methods control the Lyapunov exponents. Compared to the previous methods, the positive Lyapunov exponents are used to reconstruct a hyperchaotic system.

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