Why are chaotic attractors rare in multistable systems?

We show that chaotic attractors are rarely found in multistable dissipative systems close to the conservative limit. As we approach this limit, the parameter intervals for the existence of chaotic attractors as well as the volume of their basins of attraction in a bounded region of the state space shrink very rapidly. An important role in the disappearance of these attractors is played by particular points in parameter space, namely, the double crises accompanied by a basin boundary metamorphosis. Scaling relations between successive double crises are presented. Furthermore, along this path of double crises, we obtain scaling laws for the disappearance of chaotic attractors and their basins of attraction.     

A geometric theory of chaotic phase synchronization

A rigorous mathematical treatment of chaotic phase synchronization is still lacking, although it has been observed in many numerical and experimental studies. In this article we address the extension of results on phase synchronization in periodic oscillators to systems with phase coherent chaotic attractors with small phase diffusion. As models of such systems we consider special flows over diffeomorphisms in which the neutral direction is periodically perturbed. A generalization of the Averaging Theorem for periodic systems is used to extend Kuramoto's geometric theory of phase locking in periodically forced limit cycle oscillators to this class of systems. This approach results in reduced equations describing the dynamics of the phase difference between drive and response systems over long time intervals. The reduced equations are used to illustrate how the structure of a chaotic attractor is important in its response to a periodic perturbation, and to conclude that chaotic phase coherent systems may not always be treated as noisy periodic oscillators in this context. Although this approach is strictly justified for periodic perturbations affecting only the phase variable of a chaotic oscillator, we argue that these ideas are applicable much more generally.     

分数阶Lorenz系统的分析及电路实现

频域传递函数近似方法不仅是常用的 分数阶混沌系统相轨迹的数值分析方法之一, 而且也是设计分数阶混沌系统电路的主要方法. 应用该方法首先研究了分数阶Lorenz系统的混沌特性, 通过对Lyapunov指数图、分岔图和数值仿真分析, 发现了其较为丰富的动态特性, 即当分数阶次从0.7到0.9以步长0.1变化时, 该分数阶Lorenz系统既存在混沌特性, 又存在周期特性, 从数值分析上说明了在更低维的Lorenz系统中存在着混沌现象. 然后又基于该方法和整数阶混沌电路的设计方法, 设计了一个模拟电路实现了该分数阶Lorenz系统, 电路中的电阻和电容等数值是由系统参数和频域传递函数近似确定的. 通过示波器观测到了该分数阶Lorenz系统的混沌吸引子和周期吸引子的相轨迹图, 这些电路实验结果与数值仿真分析是一致的, 进一步从物理实现上说明了其混沌特性. 关键词： 分数阶系统 Lorenz系统 分岔分析 电路实现     

Complex dynamic behaviors in hyperbolic-type memristor-based cellular neural network

This paper presents a new hyperbolic-type memristor model,whose frequency-dependent pinched hysteresis loops and equivalent circuit are tested by numerical simulations and analog integrated operational amplifier circuits.Based on the hyperbolic-type memristor model,we design a cellular neural network(CNN)with 3-neurons,whose characteristics are analyzed by bifurcations,basins of attraction,complexity analysis,and circuit simulations.We find that the memristive CNN can exhibit some complex dynamic behaviors,including multi-equilibrium points,state-dependent bifurcations,various coexisting chaotic and periodic attractors,and offset of the positions of attractors.By calculating the complexity of the memristor-based CNN system through the spectral entropy(SE)analysis,it can be seen that the complexity curve is consistent with the Lyapunov exponent spectrum,i.e.,when the system is in the chaotic state,its SE complexity is higher,while when the system is in the periodic state,its SE complexity is lower.Finally,the realizability and chaotic characteristics of the memristive CNN system are verified by an analog circuit simulation experiment.     

Cycling chaotic attractors in two models for dynamics with invariant subspaces

Nonergodic attractors can robustly appear in symmetric systems as structurally stable cycles between saddle-type invariant sets. These saddles may be chaotic giving rise to "cycling chaos." The robustness of such attractors appears by virtue of the fact that the connections are robust within some invariant subspace. We consider two previously studied examples and examine these in detail for a number of effects: (i) presence of internal symmetries within the chaotic saddles, (ii) phase-resetting, where only a limited set of connecting trajectories between saddles are possible, and (iii) multistability of periodic orbits near bifurcation to cycling attractors. The first model consists of three cyclically coupled Lorenz equations and was investigated first by Dellnitz et al. [Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 1243-1247 (1995)]. We show that one can find a "false phase-resetting" effect here due to the presence of a skew product structure for the dynamics in an invariant subspace; we verify this by considering a more general bi-directional coupling. The presence of internal symmetries of the chaotic saddles means that the set of connections can never be clean in this system, that is, there will always be transversely repelling orbits within the saddles that are transversely attracting on average. Nonetheless we argue that "anomalous connections" are rare. The second model we consider is an approximate return mapping near the stable manifold of a saddle in a cycling attractor from a magnetoconvection problem previously investigated by two of the authors. Near resonance, we show that the model genuinely is phase-resetting, and there are indeed stable periodic orbits of arbitrarily long period close to resonance, as previously conjectured. We examine the set of nearby periodic orbits in both parameter and phase space and show that their structure appears to be much more complicated than previously suspected. In particular, the basins of attraction of the periodic orbits appear to be pseudo-riddled in the terminology of Lai [Physica D 150, 1-13 (2001)].     

Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.     

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.     

Generating multi-layer nested chaotic attractor and its FPGA implementation

Complex chaotic sequences are widely employed in real world, so obtaining more complex sequences have received highly interest. For enhancing the complexity of chaotic sequences, a common approach is increasing the scroll-number of attractors. In this paper, a novel method to control system for generating multi-layer nested chaotic attractors is proposed.At first, a piecewise(PW) function, namely quadratic staircase function, is designed. Unlike pulse signals, each level-logic of this function is square constant, and it is easy to realize. Then, by introducing the PW functions to a modified Chua's system with cubic nonlinear terms, the system can generate multi-layer nested Chua's attractors. The dynamical properties of the system are numerically investigated. Finally, the hardware implementation of the chaotic system is used FPGA chip.Experimental results show that theoretical analysis and numerical simulation are right. This chaotic oscillator consuming low power and utilization less resources is suitable for real applications.     

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.     

非线性电路通向混沌的演化过程

给出了四阶非线性电路通向复杂性的两种演化模式,指出这两种模式与三个共存的平衡点有关.在第一种模式中,不稳定的平衡点由Hopf分岔导致了稳定的周期运动,经过倍周期分岔通向混沌,其所有的吸引子都保持对称结构;而在第二种模式中,另两个平衡点由Hopf分岔产生相互对称的极限环,并分别导致了两个混沌吸引子,其分岔过程步调一致,而且所有的吸引子都相互对称.随着参数的变化,这两个混沌吸引子相互作用形成一个扩大的混沌吸引子,导致与第一种分岔模式中定性一致的混沌运动.     

Cycles homoclinic to chaotic sets; robustness and resonance

For dynamical systems possessing invariant subspaces one can have a robust homoclinic cycle to a chaotic set. If such a cycle is stable, it manifests itself as long periods of quiescent chaotic behaviour interrupted by sudden transient 'bursts'. The time between the transients increases as the trajectory approaches the cycle. This behavior for a cycle connecting symmetrically related chaotic sets has been called 'cycling chaos' by Dellnitz et al. [IEEE Trans. Circ. Sys. I 42, 821-823 (1995)]. We characterise such cycles and their stability by means of normal Lyapunov exponents. We find persistence of states that are not Lyapunov stable but still attracting, and also states that are approximately periodic. For systems possessing a skew-product structure (such as naturally arises in chaotically forced systems) we show that the asymptotic stability and the attractivity of the cycle depends in a crucial way on what we call the footprint of the cycle. This is the spectrum of Lyapunov exponents of the chaotic invariant set in the expanding and contracting directions of the cycle. Numerical simulations and calculations for an example system of a homoclinic cycle parametrically forced by a Rossler attractor are presented; here we observe the creation of nearby chaotic attractors at resonance of transverse Lyapunov exponents. (c) 1997 American Institute of Physics.     

一类新的边界激变现象:混沌的边界激变

混沌吸引子的激变是一类普遍现象.借助于广义胞映射图论(generalized cell mapping digraph)方法发现了嵌入在分形吸引域边界内的混沌鞍,这个混沌鞍由于碰撞混沌吸引子导致混沌吸引子完全突然消失,是一类新的边界激变现象,称为混沌的边界激变.可以证明混沌的边界激变是由于混沌吸引子与分形吸引域边界上的混沌鞍相碰撞产生的,在这种情况下,当系统参数通过激变临界值时,混沌吸引子连同它的吸引域突然消失,同时这个混沌鞍也突然增大 关键词： 广义胞映射 有向图 激变 混沌鞍     

Periodic window arising in the parameter space of an impact oscillator

In the bi-dimensional parameter space of an impact-pair system, shrimp-shaped periodic windows are embedded in chaotic regions. We show that a weak periodic forcing generates new periodic windows near the unperturbed one with its shape and periodicity. Thus, the new periodic windows are parameter range extensions for which the controlled periodic oscillations substitute the chaotic oscillations. We identify periodic and chaotic attractors by their largest Lyapunov exponents.     

Generating multi-scroll chaotic attractors by thresholding

This Letter proposes a novel thresholding approach for creating multi-scroll chaotic attractors. The general jerk circuit and Chua's circuit with sine nonlinearity are then used as two representative examples to show the working principle of this method. The controlled jerk circuit can generate various limit cycles and multi-scroll chaotic attractors by tuning the thresholds and the width of inner threshold plateau. The dynamic mechanism of threshold control is further explored by analyzing the system dynamical behaviors. In particular, this approach is effective and easy to be implemented since we only need to monitor the threshold variables or their functions and then reset them if they exceed the desired thresholds. Furthermore, two simple block circuit diagrams with threshold controllers are designed for the implementations of 1, 2, 3-scroll chaotic attractors. It indicates the potential engineering applications for various chaos-based information systems.     

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.     

非线性动力系统分岔图中的暗线

基于非线性动力系统混沌运动的回归特性，构造了一种对分岔图中穿过混沌区的暗线进行研究的数值回归算法。运用该算法求得抛物线映射的暗线，并与通过暗线方程精确求得的暗线进行比较，验证了算法的有效性。对Brussel振子系统和分段线性单级齿轮动力系统的暗线进行了研究。通过对非线性动力系统分岔图中暗线的研究，由其切点可以得到嵌在混沌区中的周期窗口，由其交点可以得到混沌吸引子的激变点。研究结果表明该算法有助于分析系统的动力学行为和控制混沌运动。     

The coupled dynamics of two particles with different limit sets

We consider a system of two coupled particles evolving in a periodic and spatially symmetric potential under the influence of external driving and damping. The particles are driven individually in such a way that in the uncoupled regime, one particle evolves on a chaotic attractor, while the other evolves on regular periodic attractors. Notably, only the latter supports coherent particle transport. The influence of the coupling between the particles is explored, and in particular how it relates to the emergence of a directed current. We show that increasing the (weak) coupling strength subdues the current in a process, which in phase-space, is related to a merging crisis of attractors forming one large chaotic attractor in phase-space. Further, we demonstrate that complete current suppression coincides with a chaos-hyperchaos transition.     

Complexity of Dynamics as Variability of Predictability

We construct a complexity measure from first principles, as an average over the “obstruction against prediction” of some observable that can be chosen by the observer. Our measure evaluates the variability of the predictability for characteristic system behaviors, which we extract by means of the thermodynamic formalism. Using theoretical and experimental applications, we show that “complex” and “chaotic” are different notions of perception. In comparison to other proposed measures of complexity, our measure is easily computable, non-divergent for the classical 1-d dynamical systems, and has properties of non-overuniversality. The measure can also be computed for higher-dimensional and experimental systems, including systems composed of different attractors. Moreover, the results of the computations made for classical 1-d dynamical systems imply that it is not the nonhyperbolicity, but the existence of a continuum of characteristic system length scales, that is at the heart of complexity.     

Photonic integrated device for chaos applications in communications

A novel photonic monolithic integrated device consisting of a distributed feedback laser, a passive resonator, and active elements that control the optical feedback properties has been designed, fabricated, and evaluated as a compact potential chaotic emitter in optical communications. Under diverse operating parameters, the device behaves in different modes providing stable solutions, periodic states, and broadband chaotic dynamics. Chaos data analysis is performed in order to quantify the complexity and chaoticity of the experimental reconstructed attractors by applying nonlinear noise filtering.