Generating one-,two-,three-and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system

A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincare maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.

Local Lyapunov Exponents and characteristics of fixed/periodic points embedded within a chaotic attractor

A chaotic dynamical system is characterized by a positive averaged exponential separation of two neighboring tra- jectories over a chaotic attractor. Knowledge of the Largest Lyapunov Exponent λ1 of a dynamical system over a bounded attractor is necessary and sufficient for determining whether it is chaotic (λ1>0) or not (λ1≤0). We intended in this work to elaborate the connection between Local Lyapunov Exponents and the Largest Lyapunov Exponent where an alternative method to calculate λ1 has emerged. Finally, we investigated some characteristics of the fixed points and periodic orbits embedded within a chaotic attractor which led to the conclusion of the existence of chaotic attractors that may not embed in any fixed point or periodic orbit within it.

A novel one equilibrium hyper-chaotic system generated upon Lü attractor

By introducing an additional state feedback into a three-dimensional autonomous chaotic attractor Lü system, this paper presents a novel four-dimensional continuous autonomous hyper-chaotic system which has only one equilibrium. There are only 8 terms in all four equations of the new hyper-chaotic system, which may be less than any other four-dimensional continuous autonomous hyper-chaotic systems generated by three-dimensional （3D） continuous autonomous chaotic systems. The hyper-chaotic system undergoes Hopf bifurcation when parameter c varies, and becomes the 3D modified Lü system when parameter k varies. Although the hyper-chaotic system does not undergo Hopf bifurcation when parameter k varies, many dynamic behaviours such as periodic attractor, quasi periodic attractor, chaotic attractor and hyper-chaotic attractor can be observed. A circuit is also designed when parameter k varies and the results of the circuit experiment are in good agreement with those of simulation.

New prediction of chaotic time series based on local Lyapunov exponent

A new method of predicting chaotic time series is presented based on a local Lyapunov exponent, by quantitatively measuring the exponential rate of separation or attraction of two infinitely close trajectories in state space. After recon- structing state space from one-dimensional chaotic time series, neighboring multiple-state vectors of the predicting point are selected to deduce the prediction formula by using the definition of the locaI Lyapunov exponent. Numerical simulations are carded out to test its effectiveness and verify its higher precision over two older methods. The effects of the number of referential state vectors and added noise on forecasting accuracy are also studied numerically.

Existence of hidden attractors in nonlinear hydro-turbine governing systems and its stability analysis

This work studies the stability and hidden dynamics of the nonlinear hydro-turbine governing system with an output limiting link,and propose a new six-dimensional system,which exhibits some hidden attractors.The parameter switching algorithm is used to numerically study the dynamic behaviors of the system.Moreover,it is investigated that for some parameters the system with a stable equilibrium point can generate strange hidden attractors.A self-excited attractor with the change of its parameters is also recognized.In addition,numerical simulations are carried out to analyze the dynamic behaviors of the proposed system by using the Lyapunov exponent spectra,Lyapunov dimensions,bifurcation diagrams,phase space orbits,and basins of attraction.Consequently,the findings in this work show that the basins of hidden attractors are tiny for which the standard computational procedure for localization is unavailable.These simulation results are conducive to better understanding of hidden chaotic attractors in higher-dimensional dynamical systems,and are also of great significance in revealing chaotic oscillations such as uncontrolled speed adjustment in the operation of hydropower station due to small changes of initial values.

Quantifying local predictability of the Lorenz system using the nonlinear local Lyapunov exponent

非线性局部Lyapunov指数(NLLE)可以用来度量混沌系统的局地可预报性。本文基于NLLE方法研究了Lorenz吸引子在相空间上的局地可预报性的空间分布特征。结果表明,在吸引子两翼的内、外边缘的局地可预报性期限较高,而吸引子中部地区的局地可预报性期限则较低。然而,局地可预报性期限的分布却没有呈现有组织的均一结构,相邻两点的局地可预报性期限可能差别很大。局地可预报性的来源被认为与吸引子上的局地动力学有关,由所在位置和在当前状态的持续时间决定。

Generation of on-off intermittency based on RSssler chaotic system

In this paper, one-state on-off intermittency and two-state on-off intermittency are generated in two five- dimensional continuum systems respectively. In each system, a two-dimensional subsystem is driven by the Rossler chaotic system. The parameter conditions under which the on-off intermittency occurs are discussed in detail. The statistical property of the intermittency is investigated. It is shown that the distribution of the laminar phase duration time follows a power law with an exponent of -3/2, which is a signature of on-off intermittency. Moreover, the phenomenon of intermingled basins is observed when attractors in the two symmetric invariant subspaces are stable. We provide an effective way to generate on-off intermittency based on a chaotic system, which is important for application and theoretical study.

A new four-dimensional chaotic system

A new four-dimensional chaotic system with a linear term and a 3-term cross product is reported. Some interesting figures of the system corresponding different parameters show rich dynamical structures.

Forming and implementing a hyperchaotic system with rich dynamics

A simple three-dimensional （3D） autonomous chaotic system is extended to four-dimensions so as to generate richer nonlinear dynamics. The new system not only inherits the dynamical characteristics of its parental 3D system but also exhibits many new and complex dynamics, including assembled 1-scroll, 2-scroll and 4-scroll attractors, as well as hyperchaotic attractors, by simply tuning a single system parameter. Lyapunov exponents and bifurcation diagrams are obtained via numerical simulations to further justify the existences of chaos and hyperchaos. Finally, an electronic circuit is constructed to implement the system, with experimental and simulation results presented and compared for demonstration and verification.

THE MATRIC ALGORITHM OF LYAPUNOV EXPONENT FOR THE EXPERIMENTAL DATA OBTAINED IN DYNAMIC ANALYSIS

The Lyapunov exponent is important quantitative index for describing chaotic attractors. Since Wolf put up the trajectory algorithm to Lyapunov exponent in 1985, how to calculate the Lyapunov exponent with accuracy has become a very important question. Based on the theoretical algorithm of Zuo Binwu, the matric algorithm of Lyapunov exponent is given, and the results with the results of Wolf's algorithm are compared. The calculating results validate that the matric algorithm has sufficient accuracy, and the relationship between the character of attractor and the value of Lyapunov exponent is studied in this paper. The corresponding conclusions are given in this paper.

Theoretic and Numerical Study of a New Chaotic System

This paper introduced a new three-dimensional continuous quadratic autonomous chaotic system, modified from the Lorenz system, in which each equation contains a single quadratic cross-product term, which is different from the Lorenz system and other existing systems. Basic properties of the new system are analyzed by means of Lyapunov exponent spectrum, Poincaré mapping, fractal dimension, power spectrum and chaotic behaviors. Furthermore, the forming mechanism of its compound structure obtained by merging together two simple attractors after performing one mirror operation has been investigated by detailed nu-merical as well as theoretical analysis. Analysis results show that this system has complex dynamics with some interesting characteristics.

A SPECTRUM OF LYAPUNOV EXPONENTS OBTAINED FROM A CHAOTIC TIME SERIES

A complete spectrum of Lyapunov exponents (LEs) is obtained from 1970— 1985 daily mean pressure measurements at Shanghai by means of a correlation matrix analysis technique and it is found that there exist LEs≥0, and <0. with their sum <zero (∑λ_1<0), thus showing the evolution of the climate-weather system represented by the series to be chaotic. The sum of positive LE is known to represent the bodily divergence of the system and the sum of these positive LEs is theoretically found to be Kolmogorov entropy of the system. This paper shows that with the time lag τ=5, the parameter m=2 and the dimensionality d_M=9, the sum of the positive LEs sum fromλ_i>0λ_i=K=0.110405 whereupon T=1 /K =9 is obtained as the predictable time scale, a result close to that acquired by the dynamic-statistical approach in early days and also in agreement with that present by the authors themselves(1991).

Method for removing interference in chaotic signals based on the Lyapunov exponent

A new method based on the phenomenon of synchronization and the properties of chaos is proposed to reduce interference in the transferred chaotic signals of synchronized systems. The interference is considered as a series of small deviations from the original clean trajectory in the phase space. By means of our special design, these small deviations can be estimated using positive Lyapunov exponents, and removed from interfered chaotic signals. Application is illustrated for the Lorenz attractor, and numerical computing demonstrates that the method is effective in removing typical external interference.

A New Four-Wing Chaotic System and Its Unified Generalized Projective Synchronization

In this paper,some basic properties of a new four-dimensional(4 D)continuous autonomous chaotic system,in which each equation contains a cubic cross-product term,are further analyzed.The new system has 9 equilibria displaying graceful symmetry with respect to the origin and coordinate planes,and the stability of them are discussed.Then detailed bifurcation analysis is given to demonstrate the evolution processes of the system.Numerical simulations show that the system evolves chaotic motions through period-doubling bifurcation or intermittence chaos while the system parameters vary.We design a new scheme of generalized projective synchronization,so-called unified generalized projective synchronization,whose response signal synchronizes with the linear combination of drive signal.The design has the advantages of containing complete synchronization,anti-synchronization and disorder synchronization over the usual generalized projective synchronization,such that it can provide greater security in secure communication.Based on Lyapunov stability theorem,some sufficient conditions for the new synchronization are inferred.Numerical simulations demonstrate the effectiveness and feasibility of the method by employing the four-wing chaotic system.

一个大Lyapunov指数四维超混沌系统分析及同步

在一个三维混沌系统的基础上构造了一个有大Lyapunov指数的四维超混沌系统,分析了该系统的耗散性和平衡点的稳定性。在所有系统参数中,改变某个参数而固定其余参数,依次计算和分析了系统各个参数的变化对系统Lyapunov指数谱的影响,从Lyapunov指数谱随各参数的变化判断系统作混沌运动和超混沌运动的参数变化区间。然后用相图研究了系统随系统参数变化时的运动规律。最后设计了一种线性反馈控制器实现了该超混沌系统的混沌同步,结果表明该方法正确有效。该系统在混沌和超混沌运动参数区间表现出的混沌和超混沌特性可以给基于混沌和超混沌理论的工程应用提供更多选择。所得结果为该超混沌系统在混沌保密通信中的应用提供了理论参考。

Chaotic characteristics and attractor evolution of friction noise during friction process

Friction experiments are conducted on a ring-on-disk tribometer, and friction noise produced during the friction process is extracted by a microphone. The phase trajectory and chaotic parameters of friction noise are obtained by phase-space reconstruction, and its attractor evolution is analyzed. The results indicate that the friction noise is chaotic because the largest Lyapunov exponent is positive. The phase trajectory of the friction noise follows a "convergence-stability-divergence" pattern during the friction process. The friction noise attractor begins forming in the running-in process, and the correlation dimension D increases gradually. In the stable process, the attractor remains steady, and D is stable. In the last step of the process, the attractor gradually disappears, and D decreases. The friction noise attractor is a chaotic attractor. Knowledge of the dynamic evolution of this attractor can help identify wear state changes from the running-in process to the steady and increasing friction processes.

DYNAMICAL ANALYSIS OF A 3-D CHAOTIC SYSTEM WITH ONLY TWO QUADRATIC NONLINEARITIES

纸与仅仅二二次的非线性报导一个三维的二次的自治混乱系统的动态学习，它是一种特殊情况所谓结合 L ü系统。这个系统的基本性质借助于 Lyapunov 代表光谱被分析；分叉图。分析证明系统与有几个周期的区域的一些有趣的在特征有复杂动力学，但是他们中的每有相当不同的周期的轨道。

Chaotic Synchronization for Dynamical System Constructed on Star Network

Using an example, it was analytically proved that the dynamical systems constructed on a star coupled network is more easy to be synchronized than that in coupled map lattices (CML). The result shows that the synchronization in small-world networks may be realized more easily. Key words Lyapunov exponent - chaotic synchronization - small-world networks MSC 2000 37D45 Project supported by the National Natural Science Foundation of China(Grant No. 10171061)

A new fractional order 6D chaotic model:Study of model dynamics,system structure graph,electronic circuit realization and fractional control

A new fractional 6D chaotic model is constructed in this paper.The new fractional 6D chaotic model has six positive parameters plus the fractional order with eight nonlinear terms.The complicated chaotic dy-namics of the new fractional 6D model is presented and analyzed.The basic properties of this model are studied and its chaotic attractors,dissipative feature,symmetry,equilibrium points,Lyapunov Exponents are investigated.The new dynamics of the 6D fractional model is numerically simulated using Matlab software.In addition,utilizing the graph theory tools certain structural characteristics are calculated.An electrical circuit is built to implement the new 5.4 fractional order 6D model.Finally,an active fractional order controller is proposed to control the new model at different fractional orders.The chaos of the new model is very useful and can be used to produce random keys for data encryption.