A novel 3D autonomous system with different multilayer chaotic attractors

This Letter proposes a novel three-dimensional autonomous system which has complex chaotic dynamics behaviors and gives analysis of novel system. More importantly, the novel system can generate three-layer chaotic attractor, four-layer chaotic attractor, five-layer chaotic attractor, multilayer chaotic attractor by choosing different parameters and initial condition. We analyze the new system by means of phase portraits, Lyapunov exponent spectrum, fractional dimension, bifurcation diagram and Poincaré maps of the system. The three-dimensional autonomous system is totally different from the well-known systems in previous work. The new multilayer chaotic attractors are also worth causing attention.     

The analysis of a novel 3-D autonomous system and circuit implementation

This Letter presents a new three-dimensional autonomous system with four quadratic terms. The system with five equilibrium points has complex chaotic dynamics behaviors. It can generate many different single chaotic attractors and double coexisting chaotic attractors over a large range of parameters. We observe that these chaotic attractors were rarely reported in previous work. The complex dynamical behaviors of the system are further investigated by means of phase portraits, Lyapunov exponents spectrum, Lyapunov dimension, dissipativeness of system, bifurcation diagram and Poincaré map. The physical circuit experimental results of the chaotic attractors show agreement with numerical simulations. More importantly, the analysis of frequency spectrum shows that the novel system has a broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

A three-scroll chaotic attractor

This Letter introduces a new chaotic member to the three-dimensional smooth autonomous quadratic system family, which derived from the classical Lorenz system but exhibits a three-scroll chaotic attractor. Interestingly, the two other scrolls are symmetry related with respect to the z-axis as for the Lorenz attractor, but the third scroll of this three-scroll chaotic attractor is around the z-axis. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, Poincaré map and chaotic dynamical behaviors of the new chaotic system are investigated, either numerically or analytically. The obtained results clearly show this is a new chaotic system and deserves further detailed investigation.     

A General Response System Control Method Based on Backstepping Design for Synchronization of Continuous Scalar Chaotic Signal

A general response system control method for synchronization of continuous scalar chaotic signal is presented. The proposed canonical genera/response system can cover most of the well-known chaotic systems. Conversely, each of these chaotic systems can Mso be used to construct the genera/response system. Furthermore, a novel controller of the proposed response system is designed based on backstepping technique, with which the output of the genera/response system and the given continuous chaotic signal can synchronize perfectly. Two numerical examples are given to illustrate the effectiveness of the proposed control method.     

A Novel Adaptive Observer-Based Control Scheme for Synchronization and Suppression of a Class of Uncertain Chaotic Systems

A novel adaptive observer-based control scheme is presented for synchronization and suppression of a class of uncertain chaotic system. First, an adaptive observer based on an orthogonal neural network is designed. Subsequently, the sliding mode controllers via the proposed adaptive observer are proposed for synchronization and suppression of the uncertain chaotic systems. Theoretical analysis and numerical simulation show the effectiveness of the proposed scheme.     

The evolution of chaotic dynamics for fractional unified system

Based on reliable numerical approach, this Letter studies the chaotic behavior of the fractional unified system. The lowest orders for this system to have a complete chaotic attractor (the attractor covers the three equilibrium points of the classical unified system) at different parameter values are obtained. A striking finding is that with the increase of the parameter α of the fractional unified system from 0 to 1, the lowest order for this system to have a complete chaotic attractor monotonically decreases from 2.97 to 2.07. Because of the inherent attribute (memory effects) of fractional derivatives, this finding reveals that the chaotic behavior of fractional (classical) unified system becomes stronger and stronger when α increases from 0 to 1. Furthermore, this Letter introduces a novel measure to characterize the chaos intensity of fractional (classical) differential system.     

Image encryption with chaotically coupled chaotic maps

We present a novel secure cryptosystem for direct encryption of color images, based on chaotically coupled chaotic maps. The proposed cipher provides good confusion and diffusion properties that ensures extremely high security because of the chaotic mixing of pixels’ colors. Information is mixed and distributed over a complete image using a complex strategy that makes known plaintext attack unfeasible. The encryption algorithm guarantees the three main goals of cryptography: strong cryptographic security, short encryption/decryption time, and robustness against noise and other external disturbances. Due to the high speed, the proposed cryptosystem is suitable for application in real-time communication systems.     

Synchronization and control of chaos in coupled chaotic multimode Nd:YAG lasers

In this Letter we numerically investigate the dynamics of a system of two coupled chaotic multimode Nd:YAG lasers with two mode and three mode outputs. Unidirectional and bidirectional coupling schemes are adopted; intensity time series plots, phase space plots and synchronization plots are used for studying the dynamics. Quality of synchronization is measured using correlation index plots. It is found that for laser with two mode output bidirectional direct coupling scheme is found to be effective in achieving complete synchronization, control of chaos and amplification in output intensity. For laser with three mode output, bidirectional difference coupling scheme gives much better chaotic synchronization as compared to unidirectional difference coupling but at the cost of higher coupling strength. We also conclude that the coupling scheme and system properties play an important role in determining the type of synchronization exhibited by the system.     

Projective cluster synchronization in drive-response dynamical networks

In this paper, we study the projective cluster synchronization in a drive-response dynamical network with 1+N coupled partially linear chaotic systems. Because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable, pinning control ideas are adopted to direct the different scaling factors onto the desired values. It is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results.     

Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors

In this paper we study different aspects of the paradigmatic Rössler model. We perform a detailed study of the local and global bifurcations of codimension one and two of limit cycles. This provides us a global idea of the three-parametric evolution of the system. We also study the regions of parameters where we may expect a chaotic behavior by the use of different Chaos Indicators. The combination of the different techniques gives an idea of the different routes to chaos and the different kinds of chaotic attractors we may found in this system.     

A new five-term simple chaotic attractor

A new chaotic attractor is presented with only five terms in three simple differential equations having fewer terms and simpler than those of existing seven-term or six-term chaotic attractors. Basic dynamical properties of the new attractor are demonstrated in terms of equilibria, Jacobian matrices, non-generalized Lorenz systems, Lyapunov exponents, a dissipative system, a chaotic waveform in time domain, a continuous frequency spectrum, Poincaré maps, bifurcations and forming mechanisms of its compound structures.     

Analyses of Optimal Embedding Dimension and Delay for Local Linear Prediction Model

In the reconstructed phase space, a novel local linear prediction model is proposed to predict chaotic time series. The parameters of the proposed model take the values that are different from those of the phase space reconstruction. We propose a criterion based on prediction error to determine the optimal parameters of the proposed model. The simulation results show that the proposed model can effectively make one-step and multistep prediction for chaotic time series, and the one-step and multi-step prediction accuracy of the proposed model is superior to that of the traditional local linear prediction.     

Comparison of feedback control methods for a hyperchaotic Lorenz system

More and more attention has been payed to the hyperchaotic system for the huge potential applications of hyperchaotic system such as secure communication and more complex structure than chaotic system. So at present the controlling of the hyperchaotic system simply and effectively is a frontier topic of nonlinear science. In this Letter, for the latest hyperchaotic Lorenz system, four feedback control methods were studied with analytic solution and necessary numerical simulations. It is found that the enhancing feedback control approach is the best choice of the given four feedback control methods for its relatively simple external inputs and relatively small necessary feedback coefficient after comparison. The conclusion is a helpful for the choice of control methods of any other chaotic and hyperchaotic systems.     

A quasi-crisis in a quasi-dissipative system

A system concatenated by two area-preserving maps may be addressed as “quasi-dissipative", since such a system can display dissipative behaviors. This is due to noninvertibility induced by discontinuity in the system function. In such a system, the image set of the discontinuous border forms a chaotic quasi-attractor. At a critical control parameter value the quasi-attractor suddenly vanishes. The chaotic iterations escape, via a leaking hole, to an emergent period-8 elliptic island. The hole is the intersection of the chaotic quasi-attractor and the period-8 island. The chaotic quasi-attractor thus changes to chaotic quasi-transients. The scaling behavior that drives the quasi-crisis has been investigated numerically. Received 29 May 2001 and Received in final form 6 November 2001     

Grazing-induced crises in hybrid dynamical systems

In hybrid dynamical systems including both continuous and discrete components, an interplay between a continuous trajectory and a discontinuity boundary can trigger a sudden qualitative change in the system dynamics. Grazing phenomena, which occur when a continuous trajectory hits a boundary tangentially, are well known as a representative of such phenomena. We demonstrate that a grazing phenomenon of a chaotic attractor can result in its sudden disappearance and initiate chaotic transients. The mechanism of this grazing-induced crisis is revealed in an illustrative example. Furthermore, we derive a formula to obtain the critical exponent of the power law on the mean duration of chaotic transients.     

Exponential decay and scaling laws in noisy chaotic scattering

In this Letter we present a numerical study of the effect of noise on a chaotic scattering problem in open Hamiltonian systems. We use the second order Heun method for stochastic differential equations in order to integrate the equations of motion of a two-dimensional flow with additive white Gaussian noise. We use as a prototype model the paradigmatic Hénon-Heiles Hamiltonian with weak dissipation which is a well-known example of a system with escapes. We study the behavior of the scattering particles in the scattering region, finding an abrupt change of the decay law from algebraic to exponential due to the effects of noise. Moreover, we find a linear scaling law between the coefficient of the exponential law and the intensity of noise. These results are of a general nature in the sense that the same behavior appears when we choose as a model a two-dimensional discrete map with uniform noise (bounded in a particular interval and zero otherwise), showing the validity of the algorithm used. We believe the results of this work be useful for a better understanding of chaotic scattering in more realistic situations, where noise is presented.     

Chaotic time series prediction: From one to another

In this Letter, a new local linear prediction model is proposed to predict a chaotic time series of a component x(t) by using the chaotic time series of another component y(t) in the same system with x(t). Our approach is based on the phase space reconstruction coming from the Takens embedding theorem. To illustrate our results, we present an example of Lorenz system and compare with the performance of the original local linear prediction model.     

Experimental Confirmation of a Modified Lorenz System

We experimentally demonstrate the butterfly-shaped chaotic attractor we have proposed before lint. J. Nonlin. Sci. Numerical Simulation 7 （2006） 187]. Some basic dynamical properties and chaotic behaviour of this new butterfly attractor are studied and they are in agreement with the results of our theoretical analysis. Moreover, the proposed system is experimental demonstrated.     

Investigation of a Unified Chaotic System and Its Synchronization by Simulations*

We investigate a unified chaotic system and its synchronization including feedback synchronization and adaptive synchronization by numerical simulations. We propose a new dynamical quantity denoted by K, which connects adaptive synchronization and feedback synchronization, to analyze synchronization schemes. We find that K can estimate the smallest coupling strength for a unified chaotic system whether it is complete feedback or one-sided feedback. Based on the previous work, we also give a new dynamical method to compute the leading Lyapunov exponent.     

Fuzzy adaptive synchronization of uncertain chaotic systems via delayed feedback control

Based on the T-S fuzzy model and the delayed feedback control (DFC) scheme, this Letter presents a robust synchronization strategy for a class of chaotic system with unknown parameters and disturbances. Being the response system, the designed robust observer can adaptively track the drive system globally. The T-S fuzzy model of the 4D chaotic system (Lorenz-Stenflo) is developed as an example for illustration. Numerical simulations are shown to verify the results.