Ssler System Research Articles

Projective Synchronization of The Modified Fractional-Order Hyperchaotic R\"{o}ssler System and Its Application in Secure Communication

In this paper, we propose a new approach to investigate the chaos projective synchronization of the modified fractional-order hyperchaotic Rossler system and its application in secure communication. The proposed communication system consists of four main elements including: modulation, master system, slave system and demodulation. The main idea of this approach is to inject the bounded or unbounded message into one of the parameters of the proposed system using the exponential function. However, the way of injecting the message in the modulation parameter must not remove the hyperchaotic character of the signal sent to the slave system. The slave system adaptively synchronizes with the master system, and the information signal can be recovered. Based on the Lyapunov stability theory, an adaptation laws and adaptive control are designed to achieve projection synchronization of the modified system. Numerical simulations are performed to show the feasibility of the proposed secure communication scheme.

Selection of Optimal Embedding Parameters Applied to Short and Noisy Time Series from Rössler System

Throughout scientific research, the state space reconstruction that embeds a non-linear time series is the first and necessary step for characterizing and predicting the behavior of a complex system. This requires to choose appropriate values of time delay T and embedding dimension dE. Three methods are applied and discussed on nonlinear time series provided by the Rossler attractor equations set: Cao’s method, the C-C method developed by Kim et al. and the C-C-1 method developed by Cai et al. A way to fix a parameter necessary to implement the last method is given. Focus has been put on small size and/or noisy time series. The reconstruction quality is measured by using a criterion based on the transformation smoothness.

Asymmetric Bistability in the R\"{o}ssler System

Asymmetric Bistability in the R\"{o}ssler System

Analysis on stability of continuous chaotic systems

The notion of k-error exhaustive entropy was proposed,based on exhaustive entropy.It was used to measure the strength of random-like property of chaotic sequences,and its two basic properties were proved.Then the method was used to analyze the stability of random-like property of three common continuous chaotic systems,such as Lorenz system,Rssler system,and Chua's system.The simulation results show that the approach can reflect the random essence of continuous chaotic system and Chua's system is better than Lorenz system and Rssler system as the source of randomness.

Chaos synchronization of a complex network with different nodes

A method of chaos synchronization of a complex network with different nodes is proposed. The complex network is constructed by chaos systems with diverse structures. The coupling functions of the connected nodes of the complex network are identified based on Lyapunov stability theory. The Rssler system，Coullet system and Lorenz system are taken as network nodes of the complex network in the paper. Simulation results show that a stable synchronization is obtained in whole complex network. The method can achieve synchronization of the networks with nodes of any chaotic systems，and the number of the node does not affect the stability of the whole complex network，therefore，it has a universal application.

An approach to phase space reconstruction from multivariate data based on data fusion

To cope with the problem of phase space reconstruction from univariate time series and multivariate time series, the novel approach to phase space reconstruction from multivariate data based on data fusion is presented in this paper. According to Bayes estimation theory, the phase points in the same phase space reconstructed from multivariate data are fused, then an optimal fusion phase space could be determined. The approach is applied to multivariate phase space reconstructions of Lorenz system and Rssler system, respectively. Compared with the figures reconstructed from univariate datas, the information reconstructed from multivariate data includes the main characters of all univariate data and represents the comprehensive information of system attractor, which makes the phase space reconstructed more abundant. At last, the approach is applied to multivariate phase space reconstruction of oil film whirling in the rotor system. The information reconstructed includes all the characters of the system, which improves the veracity for fault diagnosis. So all the analysis further shows that the approach presented here is effective.

Synchronization of fractional-order chaotic systems with different structures

Based on stability theory of fractional-order linear systems, a novel method combining feedback control and active control is proposed for the synchronization between two different fractional-order chaotic systems. The controller is obtained, and the method can be applied to solve synchronization problems of several classes of fractional-order chaotic systems, e.g., Lorenz system, Rssler system, Chen system, Liu system, Lü system, Rssler hyperchaotic system and the new hyperchaotic system. Numerical simulation results are presented to demonstrate the effectiveness and feasibility of the proposed method.

Adaptive slide mode control for a class of chaotic systems with perturbations

Based on the adaptive slide mode control method, we investigated the robust synchronization for a class of chaotic systems with random perturbations. This approach can be applied to may chaotic systems; and the upper bound of random perturbations is not necessarily known in advance and it is estimated through an adaptive control process. Finally, two-scroll chaotic system, Lorenz system and hyperchaotic Rssler system are taken as examples to illustrate the effectiveness of the proposed method.

Projective synchronization of autonomous chaotic system based on linear separation

This paper analyses the projective synchronization in autonomous chaotic system. Base on the stability criterion of linear systems, a new approach for constructing chaotically projective synchronization is proposed. The projective synchronizations of Lorenz system, Rssler system and hyperchaotic chen's system are achieved via the linear separation method. Numerical simulations are provided for illustration and verification of the proposed method.

A novel chaotic system via combining a unified system with R?ssler system

This brief paper designs a novel chaotic system via convex-combining a unified system with Rssler system. The combined system includes a Unified-Rssler system with double parameters, a Lorenz-Rssler system and a Chen-Rssler system with single parameter. The chaotic intervals in Chen-Rssler system are determined by calculating maximum Lyapunov exponent spectrum and numerical simulation.

Synchronization for hyperchaotic Chen system and hyperchaotic R?ssler system with different structure

In this paper two different methods——active control synchronization method and adaptive synchronization method are presented to study hyperchaotic Chen system and hyperchaotic Rssler system with different structure. Two different controllers are designed for synchronizing response system and drive system. When the parameters are known in advance, active control synchronization is adopted, the method is simple without constructing Lyapunov function and the time which realizes synchronization is short, when the parameters are fully unknown, the analytical expression of synchronization controller and adaptive law of parameters is given based on Lyapunov stability theory, which makes the different systems reach to synchronization and identifies the unknown parameters. Numerical simulation shows the effectiveness and feasibility of these two methods.

Chaotic time series forecasting using online least squares support vector machine regression

A chaotic time series forecasting method based on online least squares support vector machine (LS-SVM) regression is proposed. The difference between the online LS-SVM and offline support vector machine (SVM) is that the online LS-SVM is still effective for the chaotic system with a variation of the system parameter. Four chaotic time series, namely, Chen's system, Rssler system, Hénon map an d chaotic electroencephalogram (EEG) signal, are used to evaluate the performanc e. The results verify the ability of the method in chaotic time series predictio n.

Synchronization of symmetrically nonlinear-coupled chaotic systems

The synchronization of two symmetrical nonlinear-coupled chaotic systems is discussed. A special nonlinear-coupled term is constructed by suitable separation between linear and nonlinear terms of the chaotic system. The phenomenon of stab le chaotic synchronization has been found in a certain region of the coupled str ength α=05. The stability of the synchronous state is examined by the sta bility criterion of linear systems and the conditional Lyapunov exponent. The Lo renz system and the hyper-chaotic Rssler system are treated as numerical exam ples. It is shown that the proposed method is effective for chaotic synchronizat ion of time-continued systems and hyper-chaotic systems.

Generalized synchronization of bidirectionally coupled chaos systems

The conditional entropy is used to assess the generalized synchronization of bidirectionally coupled chaotic systems. By increasing the coupled strength, the generalized synchronization appears when there is a sharp minimum of the conditional entropy. We apply this criteria to the case of the coupled Rssler-Lorenz an d Rssler-Rssler system to test our method. Numerical results show that it is very effective and robust.

Investigation of dynamical systems using tools of the theory of invariants and projective geometry

The investigation of nonlinear dynamical systems of the type $\dot{x}=P(x,y,z),\dot{y}=Q(x,y,z),\dot{z}=R(x,y,z)$ by means of reduction to some ordinary differential equations of the second order in the form $y''+a_1(x,y)y'^3+3a_2(x,y)y'^2+3a_3(x,y)y'+a_4(x,y)=0$ is done. The main backbone of this investigation was provided by the theory of invariants developed by S. Lie, R. Liouville and A. Tresse at the end of the 19th century and the projective geometry of E. Cartan. In our work two, in some sense supplementary, systems are considered: the Lorenz system $\dot{x}=\sigma (y-x), \dot{y}=rx-y-zx,\dot{z}=xy-bz $ and the R\"o\ss ler system $\dot{x}=-y-z,\dot{y}=x+ay,\dot{z}=b+xz-cz.$. The invarinats for the ordinary differential equations, which correspond to the systems mentioned abouve, are evaluated. The connection of values of the invariants with characteristics of dynamical systems is established.

Control of Chaos in the Rössler System

The control method of Ott, Grebogi and Yorke (1990) as applied to the Rössler system, a set of three-dimensional non-linear differential equations, is examined. Using numerical time series data for a single dynamical variable the method was successfully employed to control several of the unstable periodic orbits in a three-dimensional embedding of the data. The method also failed for a number of unstable periodic orbits due to difficulties in linearising about the orbit or the tangential coincidence of the stable manifold and the motion of the orbit with external parameter.