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.     

A novel memristive time–delay chaotic system without equilibrium points

Memristor and time–delay are potential candidates for constructing new systems with complex dynamics and special features. A novel time–delay system with a presence of memristive device is proposed in this work. It is worth noting that this memristive time–delay system can generate chaotic attractors although it possesses no equilibrium points. In addition, a circuitry implementation of such time–delay system has been introduced to show its feasibility.     

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.     

Multistability and coexisting transient chaos in a simple memcapacitive system

The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper. The critical parameter of stable and unstable states is calculated by identifying the eigenvalues of Jacobian matrix. Besides, complex dynamical behaviors are investigated in the system, such as coexisting attractors, hidden attractors,coexisting bifurcation modes, intermittent chaos, and multistability. From the theoretical analyses and numerical simulations, it is found that there are four different kinds of transient transition behaviors in the memcapacitive system. Finally,field programmable gate array(FPGA) is used to implement the proposed chaotic system.     

On some properties of nearly conservative dynamics of Ikeda map and its relation with the conservative case

The behavior of the well-known Ikeda map with very weak dissipation (so-called nearly conservative case) is investigated. The changes in the bifurcation structure of the parameter plane while decreasing the dissipation are revealed. It is shown that when the dissipation is very weak the system demonstrates an “intermediate” type of dynamics combining the peculiarities of conservative and dissipative dynamics. The correspondence between the trajectories in the phase space in the conservative case and the transformations of the set of initial conditions in the nearly conservative case has been obtained. The dramatic increase of the number of coexisting low-period attractors and the extraordinary growth of the transient time while the dissipation decreases have been revealed. The method of plotting a bifurcation tree for the set of initial conditions has been used to classify the existing attractors by their structure. Also it was shown that most of the coexisting attractors are destroyed by rather small external noise, and the transient time in noisy driven systems increases still more. The new method of two-parameter analysis for conservative systems was proposed.     

Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system

A novel 5-dimensional(5D) memristive chaotic system is proposed, in which multi-scroll hidden attractors and multiwing hidden attractors can be observed on different phase planes. The dynamical system has multiple lines of equilibria or no equilibrium when the system parameters are appropriately selected, and the multi-scroll hidden attractors and multi-wing hidden attractors have nothing to do with the system equilibria. Particularly, the numbers of multi-scroll hidden attractors and multi-wing hidden attractors are sensitive to the transient simulation time and the initial values. Dynamical properties of the system, such as phase plane, time series, frequency spectra, Lyapunov exponent, and Poincar′e map, are studied in detail. In addition, a state feedback controller is designed to select multiple hidden attractors within a long enough simulation time. Finally, an electronic circuit is realized in Pspice, and the experimental results are in agreement with the numerical ones.     

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.     

Enhancing Chaos Complexity of a Plasma Model through Power Input with Desirable Random Features

The present work introduces an analysis framework to comprehend the dynamics of a 3D plasma model, which has been proposed to describe the pellet injection in tokamaks. The analysis of the system reveals the existence of a complex transition from transient chaos to steady periodic behavior. Additionally, without adding any kind of forcing term or controllers, we demonstrate that the system can be changed to become a multi-stable model by injecting more power input. In this regard, we observe that increasing the power input can fluctuate the numerical solution of the system from coexisting symmetric chaotic attractors to the coexistence of infinitely many quasi-periodic attractors. Besides that, complexity analyses based on Sample entropy are conducted, and they show that boosting power input spreads the trajectory to occupy a larger range in the phase space, thus enhancing the time series to be more complex and random. Therefore, our analysis could be important to further understand the dynamics of such models, and it can demonstrate the possibility of applying this system for generating pseudorandom sequences.     

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.     

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.     

Bistability in a hyperchaotic system with a line equilibrium

A hyperchaotic system with an infinite line of equilibrium points is described. A criterion is proposed for quantifying the hyperchaos, and the position in the three-dimensional parameter space where the hyperchaos is largest is determined. In the vicinity of this point, different dynamics are observed including periodicity, quasi-periodicity, chaos, and hyperchaos. Under some conditions, the system has a unique bistable behavior, characterized by a symmetric pair of coexisting limit cycles that undergo period doubling, forming a symmetric pair of strange attractors that merge into a single symmetric chaotic attractor that then becomes hyperchaotic. The system was implemented as an electronic circuit whose behavior confirms the numerical predictions.     

The dynamics of the pendulum suspended on the forced Duffing oscillator

We investigate the dynamics of the pendulum suspended on the forced Duffing oscillator. The detailed bifurcation analysis in two parameter space (amplitude and frequency of excitation) which presents both oscillating and rotating periodic solutions of the pendulum has been performed. We identify the areas with low number of coexisting attractors in the parameter space as the coexistence of different attractors has a significant impact on the practical usage of the proposed system as a tuned mass absorber.     

Coexisting multiscroll hyperchaotic attractors generated from a novel memristive jerk system

In this paper, two kinds of novel non-ideal voltage-controlled multi-piecewise cubic nonlinearity memristors and their mathematical models are presented. By adding the memristor to the circuit of a three-dimensional jerk system, a novel memristive multiscroll hyperchaotic jerk system is established without introducing any other ordinary nonlinear functions, from which \(2N+2\)-scroll and \(2M+1\)-scroll hyperchaotic attractors are achieved. It is exciting to note that this new memristive system can produce the extreme multistability phenomenon of coexisting infinitely multiple attractors. Furthermore, the dynamical behaviours of the proposed system are analysed by phase portraits, equilibrium points, Lyapunov exponents and bifurcation diagrams. The results indicate that the system exhibits hyperchaotic, chaotic and periodic dynamics. Especially, the phenomenon of transient chaos can also be found in this memristive multiscroll system. Additionally, the MULTISIM circuit simulations and the hardware experimental results are performed to verify numerical simulations.     

Patterned chaos in a system of three coupled identical unidirectional ring lasers

Three coupled identical unidirectional B-type ring lasers are considered and the dynamics for varying coupling parameter investigated. For given system parameters like pumping, cavity damping etc. the system is characterized by three distinct types of steady states. The linear stability of these steady states is investigated in detail which are unstable beyond a critical value of the coupling parameter. For a certain range of the coupling parameter chaotic behaviour of the system is demonstrated. The dynamical variables are mapped on to a complex variable reflecting the symmetry of the system. The plot of the real vs. the imaginary part of this complex variable exhibits patterns with threefold symmetry. The largest Lyapunov exponent as well as the power spectra corresponding to the chaotic attractors are also calculated.     

A four-wing and double-wing 3D chaotic system based on sign function

Many four-wing chaotic systems have been developed based on cross product or quadratic operations. Differently, we construct a three-dimensional chaotic system generating four-wing or double-wing attractors by virtue of sign function. Dynamical properties such as equilibrium points, Poincaré map, Lyapunov exponent spectra, Hopf bifurcations and bifurcation diagrams of the system are theoretically and numerically analyzed. Results of mathematical analyses and simulation tests indicate that the proposed chaotic system can keep chaotic to generate four-wing or double-wing attractors within a large scope of parameters. The system also shows hyperchaotic behaviors in some parameter range. Besides, circuit implementation of the chaotic system is studied. That proves the system is physically realizable.     

多平衡态下簇发振荡产生机理及吸引子结构分析

以周期激励下受控Lorenz模型为例, 考察了多平衡态共存下激励频率与系统固有频率之间存在量级差距也即存在频域上的不同尺度时的耦合效应. 由于激励频率远小于系统的固有频率, 因此将整个激励项视为慢变参数, 分析随慢变参数变化下的各种分岔模式及其相应的分岔行为, 指出在一定条件下, 不同平衡点会产生Hopf分岔和fold分岔. 根据分岔条件的不同, 给出了两种典型情况下的簇发振荡, 并通过引入转换相图, 揭示了不同簇发的产生机理, 指出多平衡态和多种分岔共存不仅会导致沉寂态和激发态的多样性, 而且会使得不同沉寂态和激发态之间存在着不同的转换形式.     

The boundary-induced spiral wave drift in the complex Ginzburg–Landau equation

We numerically investigate the boundary-induced spiral wave drift in the complex Ginzburg–Landau equation. We find some novel phenomena for the spiral drifting dynamics such as the chaotic behaviors, the transient chaos and asymmetrical attractors.     

Transient transition behaviors of fractional-order simplest chaotic circuit with bi-stable locally-active memristor and its ARM-based implementation

This paper proposes a fractional-order simplest chaotic system using a bi-stable locally-active memristor. The characteristics of the memristor and transient transition behaviors of the proposed system are analyzed, and this circuit is implemented digitally using ARM-based MCU. Firstly, the mathematical model of the memristor is designed, which is nonvolatile, locally-active and bi-stable. Secondly, the asymptotical stability of the fractional-order memristive chaotic system is investigated and some sufficient conditions of the stability are obtained. Thirdly, complex dynamics of the novel system are analyzed using phase diagram, Lyapunov exponential spectrum, bifurcation diagram, basin of attractor, and coexisting bifurcation, coexisting attractors are observed. All of these results indicate that this simple system contains the abundant dynamic characteristics. Moreover, transient transition behaviors of the system are analyzed, and it is found that the behaviors of transient chaotic and transient period transition alternately occur. Finally, the hardware implementation of the fractional-order bi-stable locally-active memristive chaotic system using ARM-based STM32F750 is carried out to verify the numerical simulation results.     

Rate processes in nonlinear optical dynamics with many attractors

Kramers' 1940 paper and its successive elaborations have extensively explored the transition rate between two stable situations, that is, in the language of system dynamics, the transition between the basins of attraction of two stable fixed point attractors. In a nonequilibrium system some of the above conditions may be violated, either because one of the two fixed points is unstable, as in the case of transient phenomena, or because both fixed points are unstable, as in the case of heteroclinic chaos, or because the attractors are more complex than fixed points, as in a chaotic dynamics where two or more strange attractors coexist. Furthermore, there is recent experimental evidence of space-time complexity consisting in the alternate or simultaneous oscillation of many modes, each one with its own (possibly chaotic) dynamics. In all the above cases, coexistence of many alternative paths implies a choice, either due to noise or self-triggered by the same interacting degrees of freedom. A review of the above phenomena in the case of nonequilibrium optical systems is here presented, with the aim of stimulating theoretical investigation on these novel rate processes.     

Generating Multidirectional Variable Hidden Attractors via Newly Commensurate and Incommensurate Non-Equilibrium Fractional-Order Chaotic Systems

This article investigates a non-equilibrium chaotic system in view of commensurate and incommensurate fractional orders and with only one signum function. By varying some values of the fractional-order derivative together with some parameter values of the proposed system, different dynamical behaviors of the system are explored and discussed via several numerical simulations. This system displays complex hidden dynamics such as inversion property, chaotic bursting oscillation, multistabilty, and coexisting attractors. Besides, by means of adapting certain controlled constants, it is shown that this system possesses a three-variable offset boosting system. In conformity with the performed simulations, it also turns out that the resultant hidden attractors can be distributively ordered in a grid of three dimensions, a lattice of two dimensions, a line of one dimension, and even arbitrariness in the phase space. Through considering the Caputo fractional-order operator in all performed simulations, phase portraits in two- and three-dimensional projections, Lyapunov exponents, and the bifurcation diagrams are numerically reported in this work as beneficial exit results.