Showing papers on "Chaotic published in 2008"

Superfamily phenomena and motifs of networks induced from time series

Abstract: We introduce a transformation from time series to complex networks and then study the relative frequency of different subgraphs within that network The distribution of subgraphs can be used to distinguish between and to characterize different types of continuous dynamics: periodic, chaotic, and periodic with noise Moreover, although the general types of dynamics generate networks belonging to the same superfamily of networks, specific dynamical systems generate characteristic dynamics When applied to discrete (map-like) data this technique distinguishes chaotic maps, hyperchaotic maps, and noise data

A novel algorithm for image encryption based on mixture of chaotic maps

Abstract: Chaos-based encryption appeared recently in the early 1990s as an original application of nonlinear dynamics in the chaotic regime. In this paper, an implementation of digital image encryption scheme based on the mixture of chaotic systems is reported. The chaotic cryptography technique used in this paper is a symmetric key cryptography. In this algorithm, a typical coupled map was mixed with a one-dimensional chaotic map and used for high degree security image encryption while its speed is acceptable. The proposed algorithm is described in detail, along with its security analysis and implementation. The experimental results based on mixture of chaotic maps approves the effectiveness of the proposed method and the implementation of the algorithm. This mixture application of chaotic maps shows advantages of large key space and high-level security. The ciphertext generated by this method is the same size as the plaintext and is suitable for practical use in the secure transmission of confidential information over the Internet.

The use of NARX neural networks to predict chaotic time series

Abstract: The prediction of chaotic time series with neural networks is a traditional practical problem of dynamic systems. This paper is not intended for proposing a new model or a new methodology, but to study carefully and thoroughly several aspects of a model on which there are no enough communicated experimental data, as well as to derive conclusions that would be of interest. The recurrent neural networks (RNN) models are not only important for the forecasting of time series but also generally for the control of the dynamical system. A RNN with a sufficiently large number of neurons is a nonlinear autoregressive and moving average (NARMA) model, with "moving average" referring to the inputs. The prediction can be assimilated to identification of dynamic process. An architectural approach of RNN with embedded memory, "Nonlinear Autoregressive model process with eXogenous input" (NARX), showing promising qualities for dynamic system applications, is analyzed in this paper. The performances of the NARX model are verified for several types of chaotic or fractal time series applied as input for neural network, in relation with the number of neurons, the training algorithms and the dimensions of his embedded memory. In addition, this work has attempted to identify a way to use the classic statistical methodologies (R/S Rescaled Range analysis and Hurst exponent) to obtain new methods of improving the process efficiency of the prediction chaotic time series with NARX.

Synchronization of chaotic fractional-order systems via active sliding mode controller

Abstract: In this paper, we propose a controller based on active sliding mode theory to synchronize chaotic fractional-order systems in master–slave structure. Master and slave systems may be identical or different. Based on stability theorems in the fractional calculus, analysis of stability is performed for the proposed method. Finally, three numerical simulations (synchronizing fractional-order Lu–Lu systems, synchronizing fractional order Chen–Chen systems and synchronizing fractional-order Lu–Chen systems) are presented to show the effectiveness of the proposed controller. The simulations are implemented using two different numerical methods to solve the fractional differential equations.

A hyperchaos generated from Lorenz system

Abstract: This paper presents a four-dimension hyperchaotic Lorenz system, obtained by adding a nonlinear controller to Lorenz chaotic system. The hyperchaotic Lorenz system is studied by bifurcation diagram, Lyapunov exponents spectrum and phase diagram. Numerical simulations show that the new system’s behavior can be convergent, divergent, periodic, chaotic and hyperchaotic when the parameter varies.

Analysis of a 3D chaotic system

Abstract: A 3D nonlinear chaotic system, called the T system, is analyzed in this paper. Horseshoe chaos is investigated via the heteroclinic Shilnikov method constructing a heteroclinic connection between the saddle equilibrium points of the system. Partially numerical computations are carried out to support the analytical results.

Synchronization of moving chaotic agents.

Abstract: We consider a set of mobile agents in a two dimensional space, each one of them carrying a chaotic oscillator, and discuss the related synchronization issues under the framework of time-variant networks. In particular, we show that, as far as the time scale for the motion of the agents is much shorter than that of the associated dynamical systems, the global behavior can be characterized by a scaled all-to-all Laplacian matrix, and the synchronization conditions depend on the agent density on the plane.

Adaptive Stabilization and Synchronization for Chaotic Lur'e Systems With Time-Varying Delay

Abstract: In this paper, we propose an adaptive scheme for the stabilization and synchronization of chaotic Lur'e systems with time-varying delay. Based on the invariant principle of functional differential equations, the strength of the feedback controller is enhanced adaptively to stabilize and synchronize chaotic Lur'e systems. The derivative-constraint that the time-varying delay is required to be differentiable and its derivation is less than one can be removed by using LaSalle-Razumikhin-type theorems. The time-varying delay is allowed to be bounded without any additional constraint or unbounded with derivative-constraint. This method is analytical, rigorous and simple to implement in practice. In addition, it is quite robust against the effect of parameters uncertainty and noise. Two examples are provided to show the effectiveness of the proposed scheme. The results of the paper demonstrate the fruitfulness of the modern feedback and adaptive control theory application to the stabilization and synchronization problems for delayed chaotic systems.

A chaotic system with one saddle and two stable node-foci

Abstract: This paper reports the finding of a chaotic system with one saddle and two stable node-foci in a simple three-dimensional (3D) autonomous system. The system connects the original Lorenz system and the original Chen system and represents a transition from one to the other. The algebraical form of the chaotic attractor is very similar to the Lorenz-type systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the chaotic system has a chaotic attractor, one saddle and two stable node-foci. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions, and the compound structure of the system are analyzed and demonstrated with careful numerical simulations.

Photonic Integrated Device for Chaos Applications in Communications

Abstract: A novel photonic monolithic integrated device consisting of a distributed feedback laser, a passive resonator, and active elements that control the optical feedback properties has been designed, fabricated, and evaluated as a compact potential chaotic emitter in optical communications. Under diverse operating parameters, the device behaves in different modes providing stable solutions, periodic states, and broadband chaotic dynamics. Chaos data analysis is performed in order to quantify the complexity and chaoticity of the experimental reconstructed attractors by applying nonlinear noise filtering.

Enhancing the Bandwidth of the Optical Chaotic Signal Generated by a Semiconductor Laser With Optical Feedback

Abstract: Bandwidth enhancement of chaotic signal generated from chaotic laser by using continuous-wave optical injection is experimentally demonstrated. A distributed feedback semiconductor laser with optical feedback is employed as the chaotic laser. The bandwidth of the chaotic signal is enhanced roughly three times by optical injection into the chaotic laser compared with the bandwidth when there is no optical injection.

A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system

Abstract: This paper introduces a new 3-D quadratic autonomous system, which can generate two coexisting single-wing chaotic attractors and a pair of diagonal double-wing chaotic attractors. More importantly, the system can generate a four-wing chaotic attractor with very complicated topological structures over a large range of parameters. Some basic dynamical behaviors and the compound structure of the new 3-D system are investigated. Detailed bifurcation analysis illustrates the evolution processes of the system among two coexisting sinks, two coexisting periodic orbits, two coexisting single-wing chaotic attractors, major and minor diagonal double-wing chaotic attractors, and a four-wing chaotic attractor. Poincare-map analysis shows that the system has extremely rich dynamics. The physical existence of the four-wing chaotic attractor is verified by an electronic circuit. Finally, spectral analysis shows that the system has an extremely broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.

Synchronization of chaotic systems with delay using intermittent linear state feedback.

Abstract: This paper investigates the synchronization of coupled chaotic systems with time delay by using intermittent linear state feedback control. An exponential synchronization criterion is obtained by means of Lyapunov function and differential inequality method. Numerical simulations on the chaotic Ikeda and Lu systems are given to demonstrate the effectiveness of the theoretical results.

New Approximations and Tests of Linear Fluctuation-Response for Chaotic Nonlinear Forced-Dissipative Dynamical Systems

Abstract: We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation–dissipation theorem. Unlike the earlier work in developing fluctuation–dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation–dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forced-dissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation–dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.

Limitations of frequency domain approximation for detecting chaos in fractional order systems

Abstract: In this paper, we analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems. The number and location of equilibria, and also the stability of these points, are compared between the original system and its frequency based approximated counterpart. It is shown that the original system and its approximation are not necessarily equivalent according to the number, location and stability of the fixed points. This problem can cause erroneous results in special cases. For instance, to prove the existence of chaos in fractional order systems, numerical simulations have been largely based on frequency domain approximations, but in this paper we show that this method is not always reliable for detecting chaos. This approximation can numerically demonstrate chaos in the non-chaotic fractional order systems, or eliminate chaotic behavior from a chaotic fractional order system.

Some Applications of Fractional Calculus in Suppression of Chaotic Oscillations

Abstract: This paper presents two different stabilization methods based on the fractional-calculus theory. The first method is proposed via using the fractional differentiator, and the other is constructed based on using the fractional integrator. It has been shown that the proposed techniques can be used to suppress chaotic oscillations in 3-D chaotic systems. To show the practical capability of the methods, some experimental results on the control of chaos in chaotic circuits are presented.

Global Asymptotical Synchronization of Chaotic Lur'e Systems Using Sampled Data: A Linear Matrix Inequality Approach

Abstract: Sampled-data feedback control for master-slave synchronization schemes that consist of identical chaotic Lur'e systems is studied. Sufficient conditions for global asymptotic synchronization of such chaotic Lur'e systems are obtained using the free-weighting matrix approach and expressed in terms of linear matrix inequalities (LMIs). With the help of the LMI solvers, the sampled-data feedback control law can easily be obtained to globally asymptotically synchronize Lur'e chaotic systems. The effectiveness of the proposed method is finally illustrated via numerical simulations of chaotic Chua's circuits.

Chaotic behaviour of deterministic dissipative systems

Abstract: Introduction 1 Differential equations, maps and asymptotic behaviour 2 Transition from order to chaos 3 Numerical methods for studies of parametric dependences, bifurcations and chaos 4 Chaotic dynamics in experiments 5 Forced and coupled chemical oscillators: a case study of chaos 6 Chaos in distributed systems Appendices Bibliography Index

Synchronization of identical and non-identical 4-D chaotic systems using active control

Abstract: This paper presents chaos synchronization between two identical Lorenz–Stenflo (LS) and a new four-dimensional chaotic system (Qi systems) by using active control technique. The designed controller ensures that the state variables of the controlled chaotic slave LS and Qi systems globally synchronizes with the state variables of the master systems respectively. It is also shown that Qi system globally synchronizes with LS system under the generalized active control. The results are validated using numerical simulations.

Chaotic dynamics of the fractionally damped van der Pol equation

Abstract: This paper deals with the harmonic oscillations of a periodically excited van der Pol system where hysteresis was simulated via fractional operator representations. The fractionally damped van der Pol equation was transformed into a set of fractional integral equations and solved by a predictor–corrector method. In particular, we focus on the effect of fractional damping on the dynamic behavior. The time evolutions of the nonlinear dynamic system responses are also described using phase portraits and the Poincare map technique. Results showed that the response of the system was very sensitive to changes in the order of fractional damping. Periodic, quasi-periodic, and chaotic motions existed when the order of fractional damping was less than 1. When the order of fractional damping exceeded 1, only chaotic motion was found among all simulations in this study. Moreover, two different strange attractors were also examined.