A class of two-dimensional rational maps with self-excited and hidden attractors

This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov (Kaplan—Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.     

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.     

Landscapes of non-gradient dynamics without detailed balance: stable limit cycles and multiple attractors

Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.     

Fast convergence of spike sequences to periodic patterns in recurrent networks

The dynamical attractors are thought to underlie many biological functions of recurrent neural networks. Here we show that stable periodic spike sequences with precise timings are the attractors of the spiking dynamics of recurrent neural networks with global inhibition. Almost all spike sequences converge within a finite number of transient spikes to these attractors. The convergence is fast, especially when the global inhibition is strong. These results support the possibility that precise spatiotemporal sequences of spikes are useful for information encoding and processing in biological neural networks.     

Excitability of periodic and chaotic attractors in semiconductor lasers with optoelectronic feedback

The dynamics of a semiconductor laser with AC-coupled nonlinear optoelectronic feedback has been experimentally studied. A period doubling sequence of small periodic and chaotic attractors is observed, each of them displaying excitable features. This scenario is found also in a simplified physical model of the system, thus extending the concept of excitability, usually associated to fixed points, also to the case of higher-dimensional attractors.     

Noninvertibility and resonance in discrete-time neural networks for time-series processing

We present a computer-assisted study emphasizing certain elements of the dynamics of artificial neural networks (ANNs) used for discrete time-series processing and nonlinear system identification. The structure of the network gives rise to the possibility of multiple inverses of a phase point backward in time; this is not possible for the continuous-time system from which the time series are obtained. Using a two-dimensional illustrative model in an oscillatory regime, we study here the interaction of attractors predicted by the discrete-time ANN model (invariant circles and periodic points locked on them) with critical curves. These curves constitute a generalization of critical points for maps of the interval (in the sense of Julia-Fatou); their interaction with the model-predicted attractors plays a crucial role in the organization of the bifurcation structure and ultimately in determining the dynamic behavior predicted by the neural network.     

Reproducible sequence generation in random neural ensembles

Little is known about the conditions that neural circuits have to satisfy to generate reproducible sequences. Evidently, the genetic code cannot control all the details of the complex circuits in the brain. In this Letter, we give the conditions on the connectivity degree that lead to reproducible and robust sequences in a neural population of randomly coupled excitatory and inhibitory neurons. In contrast to the traditional theoretical view we show that the sequences do not need to be learned. In the framework proposed here just the averaged characteristics of the random circuits have to be under genetic control. We found that rhythmic sequences can be generated if random networks are in the vicinity of an excitatory-inhibitory synaptic balance. Reproducible transient sequences, on the other hand, are found far from a synaptic balance.     

Transients and asymptotics in granular phase space

The dynamics of dissipative dynamical systems can be described by the sequential appearance of two different regimes. From a given initial condition, one first observes transient behavior characterized by a high degree of contraction of volumes in phase space. This is followed by an asymptotic regime with one or several attractors into which trajectories inject after long times. There is however, no sharp crossover between these two regimes and the identification of either one depends on the precision of measurement. In order to investigate these issues, we studied the dynamics of contracting integer maps. We found out that for the cases which in the continuum limit correspond to bifurcations, transients consists of two regimes sharply separated by a crossover point which displays universal scaling with the size of the set. Moreover, their average lengths display power law dependence on the accuracy of their measurement. This behavior persists away from bifurcation but with a different scaling law. In addition, we studied deterministic diffusion on finite sets and obtained analytic expressions for the mean square displacement in the long time limit.     

Transient selection in multicellular immune networks

We analyze the dynamics of a multi-clonotype naive T-cell population competing for survival signals from antigen-presenting cells. We find that this competition provides with an efficacious selection of clonotypes, making the less able and more repetitive get extinct. We uncover the scaling principles for large systems the extinction rate obeys and calibrate the model parameters to their experimental counterparts. For the first time, we estimate the physiological values of the T-cell receptor-antigen presentation profile recognition probability and T-cell clonotypes niche overlap. We demonstrate that, while the ultimate state is a stable fixed point, sequential transients dominate the dynamics over large timescales that may span over years, if not decades, in real time. We argue that what is currently viewed as “homeostasis” is a complex sequential transient process, while being quasi-stationary in the total number of T-cells only. The discovered type of sequential transient dynamics in large random networks is a novel alternative to the stable heteroclinic channel mechanism.     

Conditional Granger causality of diffusion processes

The statistical concept of Granger causality is defined by prediction improvement, i.e. the causing time series contains unique information about the future of the caused one. Recently we proposed extending this concept to bivariate diffusion processes by defining Granger causality for each point of the state space as the Granger causality of a process obtained by local linearisation. This provides a Granger causality map, well-defined at least in the vicinity of stable fixed points of the deterministic part of the dynamics. This extension has convenient properties, but carries several important limitations. In the current paper we show how the Granger causality of diffusion processes can be further generalized, incorporating in particular the concept of conditional causality. Moreover, we demonstrate the application potential to systems with a more complex attractor structure such as limit cycles or bistability of fixed points.     

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.     

Analog-symbolic memory that tracks via reconsolidation

A fundamental part of a computational system is its memory, which is used to store and retrieve data. Classical computer memories rely on the static approach and are very different from human memories. Neural network memories are based on auto-associative attractor dynamics and thus provide a high level of pattern completion. However, they are not used in general computation since there are practically no algorithms to load an arbitrary landscape of attractors into them. In this sense neural network memory models cannot communicate well with symbolic and prior knowledge.We propose the design of a new memory based on localist attractor dynamics with reconsolidation called Reconsolidation Attractor Network (RAN). RAN combines symbolic and subsymbolic features in a very attractive way: it is based on attractors; enables pattern classification under missing data; and demonstrates dynamic reconsolidation, which is very useful for tracking changing concepts. The perception RAN enables is somewhat reminiscent of human perception due to its context sensitivity. Furthermore, it enables an immediate and clear interface with symbolic memories, including loading of attractors by means of trivial wiring, updating attractors, and retrieving them faster without waiting for full convergence. It also scales to any number of concepts. This provides a useful counterpoint to more conventional memory systems, such as random access memory and auto-associative neural networks.     

On the aging dynamics in an immune network model

Recently we have used a cellular automata model which describes the dynamics of a multi-connected network to reproduce the refractory behavior and aging effects obtained in immunization experiments performed with mice when subjected to multiple perturbations. In this paper we investigate the similarities between the aging dynamics observed in this multi-connected network and the one observed in glassy systems, by using the usual tools applied to analyze the latter. An interesting feature we show here, is that the model reproduces the biological aspects observed in the experiments during the long transient time it takes to reach the stationary state. Depending on the initial conditions, and without any perturbation, the system may reach one of a family of long-period attractors. The perturbations may drive the system from its natural attractor to other attractors of the same family. We discuss the different roles played by the small random perturbations ("noise") and by the large periodic perturbations ("immunizations").Received: 7 May 2003, Published online: 23 July 2003PACS: 87.18.Hf Spatiotemporal pattern formation in cellular populations - 87.10.+e General theory and mathematical aspects - 61.43.Fs GlassesD.A. Stariolo: Research Associate of the Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, Trieste, Italy     

Using chaotic interrogation and attractor nonlinear cross-prediction error to detect fastener preload loss in an aluminum frame

Structural health monitoring is an important field concerned with assessing the current state (or "health") of a structural system or component with regard to its ability to perform its intended function appropriately. One approach to this problem is identifying appropriate features obtained from time series vibration responses of the structure that change as structural degradation occurs. In this work, we present a novel technique adapted from the nonlinear time series prediction community whereby the structure is excited by an applied chaotic waveform, and predictive maps built between structural response attractors are used as the feature space. The structural response is measured at several points on the structure, and pairs of attractors are used to predict each other. As the dynamics of the structure change due to damage, the prediction error rises. This approach is applied to detecting the preload loss in a bolted joint in an aluminum frame structure.     

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.     

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.     

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.     

Unstable attractors induce perpetual synchronization and desynchronization

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which unstable attractors arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.     

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.