A new four-dimensional chaotic system with first Lyapunov exponent of about 22,hyperbolic curve and circular paraboloid types of equilibria and its switching synchronization by an adaptive global integral sliding mode control

This paper presents a new four-dimensional(4 D) autonomous chaotic system which has first Lyapunov exponent of about 22 and is comparatively larger than many existing three-dimensional(3 D) and 4 D chaotic systems.The proposed system exhibits hyperbolic curve and circular paraboloid types of equilibria.The system has all zero eigenvalues for a particular case of an equilibrium point.The system has various dynamical behaviors like hyperchaotic,chaotic,periodic,and quasi-periodic.The system also exhibits coexistence of attractors.Dynamical behavior of the new system is validated using circuit implementation.Further an interesting switching synchronization phenomenon is proposed for the new chaotic system.An adaptive global integral sliding mode control is designed for the switching synchronization of the proposed system.In the switching synchronization,the synchronization is shown for the switching chaotic,stable,periodic,and hybrid synchronization behaviors.Performance of the controller designed in the paper is compared with an existing controller.     

一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

对于具有隐藏吸引子的混沌系统,既有文献大多只针对整数阶系统进行分析与控制研究.基于Sprott E系统,构建了仅有一个稳定平衡点的分数阶混沌系统,通过相位图、Poincare映射和功率谱等,分析了该系统的基本动力学特征.结果显示,该系统展现出了丰富而复杂的动力学特性,且通过随阶次变化的分岔图可知,系统在不同阶次下呈现出周期运动、倍周期运动和混沌运动等状态,这些动力学特征对于保密通信等实际工程领域有重要的研究价值.针对该具有隐藏吸引子的分数阶系统,应用分数阶系统有限时间稳定性理论设计控制器,对系统进行有限时间同步控制,并通过数值仿真验证了其有效性.     

Analysis,synchronization and circuit design of a novel butterfly attractor

This research paper introduces a novel three-dimensional autonomous system, whose dynamics support periodic and chaotic butterfly attractors as certain parameters vary. A special case of this system, exhibiting reflectional symmetry, is amenable to analytical and numerical analysis. Qualitative properties of the new chaotic system are discussed in detail. Adaptive control laws are derived to achieve global chaotic synchronization of the new chaotic system with unknown parameters. Furthermore, a novel electronic circuit realization of the new chaotic system is presented, examined and realized using Orcad-PSpice program and physical components. The proposed novel butterfly chaotic attractor is very useful for the deliberate generation of chaos in applications.     

混沌系统的统一投影同步

提出一种实现混沌系统投影同步的统一方法.通过构建一个广义比例矩阵和一个合适的响应系统,建立了混沌投影同步的通用模型,依据比例矩阵的不同种类,可实现广义投影同步、错位投影同步和广义混合错位投影同步等各种类型的混沌投影同步.利用Lyapunov稳定性理论给出了相应的证明,最后以多卷波混沌系统和超混沌Qi系统为例,进行了错位投影同步和广义混合错位投影同步的数值仿真,仿真结果进一步表明了该方法的有效性. 关键词： 混沌系统 广义比例矩阵 统一投影同步     

Time-delayed chameleon: Analysis,synchronization and FPGA implementation

In this paper we report a time-delayed chameleon-like chaotic system which can belong to different families of chaotic attractors depending on the choices of parameters. Such a characteristic of self-excited and hidden chaotic flows in a simple 3D system with time delay has not been reported earlier. Dynamic analysis of the proposed time-delayed systems are analysed in time-delay space and parameter space. A novel adaptive modified functional projective lag synchronization algorithm is derived for synchronizing identical time-delayed chameleon systems with uncertain parameters. The proposed time-delayed systems and the synchronization algorithm with controllers and parameter estimates are then implemented in FPGA using hardware–software co-simulation and the results are presented.     

Construction of a New Fractional Chaotic System and Generalized Synchronization

In this paper a new dynamic system with integer and fractional order is investigated. It is shown that determining the effect of quadratic coefficients to the systematic structure can be converted to determining that of coefficients of the linear part. Under some parametric conditions, the system can produce chaotic attractors similar as Lorenz attractor. A constructive theorem is proposed for generalized synchronization related to the fractional-order chaotic system and an application of this new system is demonstrated.     

Generation and synchronization of N-scroll chaotic and hyperchaotic attractors in fourth-order systems

Based on our previous works and Lyapunov stability theory, this paper studies the generation and synchronization of N-scroll chaotic and hyperchaotic attractors in fourth-order systems. A fourth-order circuit, by introducing additional breakpoints in the modified Chua oscillator, is implemented for the study of generation and synchronization of N-scroll chaotic attractors.This confirms the consistency of theoretical calculation, numerical simulation and circuit experiment.Furthermore,we give a refined and extended study of generating and synchronizing N-scroll hyperchaotic attractors in the fourth-order MCK system and report the new theoretical result, which is verified by computer simulations.     

基于收缩映射的奇异非混沌系统同步

提出一种基于收缩映射的奇异非混沌系统同步方案．通过利用一种混沌系统驱动另一种混沌系统产生出奇异非混沌吸引子,由于奇异非混沌吸引子的Ｌｙａｐｕｎｏｖ指数为负值,因而可有效抑制混沌系统对初始状态的敏感程度．为实现两个奇异非混沌吸引子的同步,文中采用收缩映射实现混沌驱动系统的快速同步．研究表明,该方案能够快速实现同步,并且有较强的鲁棒性,易于实现,可用于混沌保密通信 关键词：     

Complex lag synchronization of two identical chaotic complex nonlinear systems

Much progress has been made in the research of synchronization for chaotic real or complex nonlinear systems. In this paper we introduce a new type of synchronization which can be studied only for chaotic complex nonlinear systems. This type of synchronization may be called complex lag synchronization (CLS). A definition of CLS is introduced and investigated for two identical chaotic complex nonlinear systems. Based on Lyapunov function a scheme is designed to achieve CLS of chaotic attractors of these systems. The effectiveness of the obtained results is illustrated by a simulation example. Numerical results are plotted to show state variables, modulus errors and phase errors of these chaotic attractors after synchronization to prove that CLS is achieved.     

Synchronization, multistability and basin crisis in coupled pendula

The synchronization dynamics of two linearly coupled pendula is studied in this paper. Based on the Lyapunov stability theory and Linear matrix inequality (LMI); some necessary and sufficient conditions for global asymptotic synchronization are derived from which an estimated threshold coupling k_th, for the on-set of full synchronization is obtained. The numerical value of k_th determined from the average energies of the systems is in good agreement with theoretical analysis. Prior to the on-set of synchronization, the boundary crisis of the chaotic attractor is identified. In the bistable states, where two asymmetric periodic attractors co-exist, it is shown that the coupled pendula can attain multistable states via a new dynamical transition—the basin crisis that occur prior to the on-set of stable synchronization. The essential feature of basin crisis is that the two co-existing attractors are destroyed while new three or more co-existing attractors of the same or different periodicity are created. In addition, the linear perturbation technique and the Routh-Hurwitz criteria are employed to investigate the stability of steady states, and clearly identify the different types of bifurcations likely to be encountered. Finally, two-parameter phase plots, show various regions of chaos, hyperchaos and periodicity.     

Anti-control and Generalized Synchronization of Chaos for a Kind of n-Dimensional Linear Differential System

This paper introduces the finding of some chaotic attractors in a kind of n-dimensional autonomous hybrid system, which is realized via applying some discontinuous state feedback to a kind of n linear differential system. And a constructive theorem is proposed for generalized synchronization related to the above chaotic hybrid systems. Examples are presented for illustrating the methods.     

基于忆阻器的多涡卷混沌系统及其脉冲同步控制

忆阻器是一种具有记忆功能和纳米级尺寸的非线性元件,作为混沌系统的非线性部分,能够提高混沌系统的信号随机性和复杂度.本文基于增广Lü系统设计了一个三维忆阻混沌系统.仅仅通过改变系统的一个参数,该系统能产生单涡巻、双涡卷和四涡巻的混沌吸引子,说明该系统具有丰富的混沌特性.首先对该忆阻混沌系统的基本动力学行为进行了理论分析和数值仿真,如平衡点稳定性、对称性,Lyapunov指数和维数,分岔图和Poincare截面等.同时,建立了模拟该忆阻混沌系统的SPICE(simulation program with integrated circuit emphasis)电路,给出了不同参数下的电路实验相图,其仿真结果与数值分析相符,从而验证了该忆阻混沌系统的混沌产生能力.由于脉冲同步只在离散时刻传递信息,能量消耗小,同步速度快,易于实现单信道传输,因而在混沌保密通信中更具有实用性.因此,本文从最大Lyapunov指数的角度实现了该忆阻混沌系统的脉冲混沌同步,数值仿真证实了忆阻混沌系统的存在性以及脉冲同步控制的可行性,为进一步研究该忆阻混沌系统在语音保密通信和信息处理中的应用提供了实验基础.     

Impulsive Synchronization of Lü Chaotic System Based on Small Impulsive Signal

In this Letter the issue of impulsive synchronization of the Lü chaotic system is developed. We propose an impulsive synchronization scheme of the Lü chaotic system including chaotic systems. Some new and sufficient conditions on varying impulsive distance are established in order to guarantee the synchronizability of the systems using the synchronization method. In particular, some simple conditions are derived in synchronizing the systems by equal impulsive distances. The proposed impulsive synchronization scheme is applied to the Lü chaotic system and the numerical example demonstrates the effectiveness of the method. The boundaries of the stable regions are also estimated.     

A new piecewise linear Chen system of fractional-order:Numerical approximation of stable attractors

In this paper we present a new version of Chen's system:a piecewise linear(PWL) Chen system of fractional-order.Via a sigmoid-like function,the discontinuous system is transformed into a continuous system.By numerical simulations,we reveal chaotic behaviors and also multistability,i.e.,the existence of small parameter windows where,for some fixed bifurcation parameter and depending on initial conditions,coexistence of stable attractors and chaotic attractors is possible.Moreover,we show that by using an algorithm to switch the bifurcation parameter,the stable attractors can be numerically approximated.     

Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control

A new kind of generalized synchronization of two chaotic systems with uncertain parameters is proposed. Based on a pragmatical asymptotical stability theorem and an assumption of equal probability for ergodic initial conditions, an adaptive control law is derived so that it can be proved strictly that the common null solution of error dynamics and of parameter dynamics is actually asymptotically stable, i.e. these two identical systems are in generalized synchronization and the estimated parameters approach the uncertain values. It is called pragmatical generalized synchronization. Finally, two numerical examples are studied for two Quantum-CNN oscillator chaotic systems to show the effectiveness of the proposed generalized synchronization strategy with a double Duffing chaotic system as a goal system.     

Three-dimensional chaotic autonomous van der pol–duffing type oscillator and its fractional-order form

In this paper, a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator is proposed. The three-dimensional autonomous VdPD oscillator is obtained by replacing the external periodic drive source of two-dimensional chaotic nonautonomous VdPD type oscillator by a direct positive feedback loop. By analyzing the stability of the equilibrium points, the existence of Hopf bifurcation is established. The dynamical properties of proposed three-dimensional autonomous VdPD type oscillator is investigated showing that for a suitable choice of the parameters, it can exhibit periodic behaviors, chaotic behaviors and coexistence between periodic and chaotic attractors. Moreover, the physical existence of the chaotic behavior and coexisting attractors found in three-dimensional proposed autonomous VdPD type oscillator is verified by using Orcard-PSpice software. A good qualitative agreement is shown between the numerical simulations and Orcard-PSpice results. In addition, fractional-order chaotic three-dimensional proposed autonomous VdPD type oscillator is studied. The lowest order of the commensurate form of this oscillator to exhibit chaotic behavior is found to be 2.979. The stability analysis of the controlled fractional-order-form of proposed three-dimensional autonomous VdPD type oscillator at its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Finally, an example of observer-based synchronization applied to unidirectional coupled identical proposed chaotic fractional-order oscillator is illustrated. It is shown that synchronization can be achieved for appropriate coupling strength.     

Dynamical analysis of a new multistable chaotic system with hidden attractor: Antimonotonicity,coexisting multiple attractors,and offset boosting

Analyzing chaotic systems with coexisting and hidden attractors has been receiving much attention recently. In this article, we analyze a four dimensional chaotic system which has a plane as the equilibrium points. Also this system is of the group of systems that have coexisting attractors. First, the system is introduced and then stability analysis, bifurcation diagram and Largest Lyapunov exponent of this system are presented as methods to analyze the multistability of the system. These methods reveal that in some ranges of the parameter, this chaotic system has three different types of coexisting attractors, chaotic, stable node and limit cycle. Some interesting dynamics properties such as reversals of period doubling bifurcation and offset boosting are also presented.     

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, Poincar\'{e} maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.     

混沌Lur’e系统的线性输出反馈同步

针对一类特殊的混沌Lur’e系统,以系统最后一状态变量作为驱动信号进行误差的线性输出反馈,提出一个同步的全局控制规律,并证明了同步的渐近稳定性.最后以蔡氏振子为例进行仿真,验证该同步方法的有效性. 关键词： 混沌同步 输出反馈 Lur’e系统 蔡氏振子     

One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamics

A novel 3D fractional-order chaotic system is proposed in this paper. And the system equations consist of nine terms including four nonlinearities. It's interesting to see that this new fractional-order chaotic system can generate one-wing, two-wing, three-wing and four-wing attractors by merely varying a single parameter. Moreover, various coexisting attractors with respect to same system parameters and different initial values and the phenomenon of transient chaos are observed in this new system. The complex dynamical properties of the presented fractional-order systems are investigated by means of theoretical analysis and numerical simulations including phase portraits, equilibrium stability, bifurcation diagram and Lyapunov exponents, chaos diagram, and so on. Furthermore, the corresponding implementation circuit is designed. The Multisim simulations and the hardware experimental results are well in accordance with numerical simulations of the same system on the Matlab platform, which verifies the correctness and feasibility of this new fractional-order chaotic system.