Robust prediction of complex spatiotemporal states through machine learning with sparse sensing

Complex spatiotemporal states arise frequently in material as well as biological systems consisting of multiple interacting units. A specific, but rather ubiquitous and interesting example is that of “chimeras”, existing in the edge between order and chaos. We use Machine Learning methods involving “observers” to predict the evolution of a system of coupled lasers, comprising turbulent chimera states and of a less chaotic biological one, of modular neuronal networks containing states that are synchronized across the networks. We demonstrated the necessity of using “observers” to improve the performance of Feed-Forward Networks in such complex systems. The robustness of the forecasting capabilities of the “Observer Feed-Forward Networks” versus the distribution of the observers, including equidistant and random, and the motion of them, including stationary and moving was also investigated. We conclude that the method has broader applicability in dynamical system context when partial dynamical information about the system is available.     

Transitions in two sinusoidally coupled Josephson junction rotators

We investigate the dynamics of two sinusoidally coupled Josephson junction rotators to provide a clear knowledge of the behaviors in different regions of the parameter space. The dynamical states are identified, and the transitions among these states are studied. The properties of the current–voltage curves are investigated. Further more, we observed the chaotic states in some regions of parameter space. We conjecture it may caused by the competition of two periodic potentials: one is the external field, another is the interacting of two particles.     

Projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks

We study projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. We relax some limitations of previous work, where projective-anticipating and projective-lag synchronization can be achieved only on two coupled chaotic systems. In this paper, we realize projective-anticipating and projective-lag synchronization on complex dynamical networks composed of a large number of interconnected components. At the same time, although previous work studied projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In this paper, the dynamics of the nodes of the complex networks are time-delayed chaotic systems without the limitation of the partial linearity. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random dynamical networks, and we find both its existence and sufficient stability conditions. The validity of the proposed method is demonstrated and verified by examining specific examples using Ikeda and Mackey-Glass systems on Erdos-Renyi networks.     

Control Meets Inference: Using Network Control to Uncover the Behaviour of Opponents

Using observational data to infer the coupling structure or parameters in dynamical systems is important in many real-world applications. In this paper, we propose a framework of strategically influencing a dynamical process that generates observations with the aim of making hidden parameters more easily inferable. More specifically, we consider a model of networked agents who exchange opinions subject to voting dynamics. Agent dynamics are subject to peer influence and to the influence of two controllers. One of these controllers is treated as passive and we presume its influence is unknown. We then consider a scenario in which the other active controller attempts to infer the passive controller’s influence from observations. Moreover, we explore how the active controller can strategically deploy its own influence to manipulate the dynamics with the aim of accelerating the convergence of its estimates of the opponent. Along with benchmark cases we propose two heuristic algorithms for designing optimal influence allocations. We establish that the proposed algorithms accelerate the inference process by strategically interacting with the network dynamics. Investigating configurations in which optimal control is deployed. We first find that agents with higher degrees and larger opponent allocations are harder to predict. Second, even factoring in strategical allocations, opponent’s influence is typically the harder to predict the more degree-heterogeneous the social network.     

On the constructive role of noise in stabilizing itinerant trajectories in chaotic dynamical systems

This work aims at studying dynamical models of neural networks, which exhibit phase transitions between states of various complexities. We use the biologically motivated KIII model, which has demonstrated excellent performance as a robust dynamical memory device. KIII is a high-dimensional dynamical system with extremely fragmented boundaries between limit cycles, tori, fixed points, and chaotic attractors. We study the role of additive noise in the development of itinerant trajectories. Noise not only stabilizes aperiodic trajectories, but there is an optimum noise level with highly itinerant behavior. We speculate that the previously found optimum classification performance of KIII as a function of the noise level, also identified as chaotic resonance, is related to chaotic itinerant oscillations among various ordered states.     

Synchronization Dynamics in Non-Normal Networks: The Trade-Off for Optimality

Synchronization is an important behavior that characterizes many natural and human made systems that are composed by several interacting units. It can be found in a broad spectrum of applications, ranging from neuroscience to power-grids, to mention a few. Such systems synchronize because of the complex set of coupling they exhibit, with the latter being modeled by complex networks. The dynamical behavior of the system and the topology of the underlying network are strongly intertwined, raising the question of the optimal architecture that makes synchronization robust. The Master Stability Function (MSF) has been proposed and extensively studied as a generic framework for tackling synchronization problems. Using this method, it has been shown that, for a class of models, synchronization in strongly directed networks is robust to external perturbations. Recent findings indicate that many real-world networks are strongly directed, being potential candidates for optimal synchronization. Moreover, many empirical networks are also strongly non-normal. Inspired by this latter fact in this work, we address the role of the non-normality in the synchronization dynamics by pointing out that standard techniques, such as the MSF, may fail to predict the stability of synchronized states. We demonstrate that, due to a transient growth that is induced by the structure’s non-normality, the system might lose synchronization, contrary to the spectral prediction. These results lead to a trade-off between non-normality and directedness that should be properly considered when designing an optimal network, enhancing the robustness of synchronization.     

Synchronization in networks with random interactions: theory and applications

Synchronization is an emergent property in networks of interacting dynamical elements. Here we review some recent results on synchronization in randomly coupled networks. Asymptotical behavior of random matrices is summarized and its impact on the synchronization of network dynamics is presented. Robert May's results on the stability of equilibrium points in linear dynamics are first extended to systems with time delayed coupling and then nonlinear systems where the synchronized dynamics can be periodic or chaotic. Finally, applications of our results to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

Synchronization rate of synchronized coupled systems

Synchronization phenomena, an emergent property in networks of interacting dynamical elements, are widely observed in nature, and have become the subject of intense research. Here we will investigate the synchronization rate problem in coupled limit-cycle and chaotic oscillators. Based on the mode decomposition method and Gershgörin's discs theorem, some sufficient conditions for synchronization of coupled systems are obtained, and a synchronization rate is then derived. Such a synchronization rate indicates that the error functions between state variables of underlying individual systems tend to zero in the exponential form as time tends to the infinity. Several numerical examples are also given to validate the theoretical results.     

Dynamical motifs: building blocks of complex dynamics in sparsely connected random networks

Spatiotemporal network dynamics is an emergent property of many complex systems that remains poorly understood. We suggest a new approach to its study based on the analysis of dynamical motifs-small subnetworks with periodic and chaotic dynamics. We simulate randomly connected neural networks and, with increasing density of connections, observe the transition from quiescence to periodic and chaotic dynamics. This transition is explained by the appearance of dynamical motifs in the structure of these networks. We also observe domination of periodic dynamics in simulations of spatially distributed networks with local connectivity and explain it by the absence of chaotic and the presence of periodic motifs in their structure.     

Complex Networks from Chaotic Time Series on Riemannian Manifold

Complex networks are important paradigms for analyzing the complex systems as they allow understanding the structural properties of systems composed of different interacting entities.In this work we propose a reliable method for constructing complex networks from chaotic time series.We first estimate the covariance matrices,then a geodesic-based distance between the covariance matrices is introduced.Consequently the network can be constructed on a Riemannian manifold where the nodes and edges correspond to the covariance matrix and geodesic-based distance,respectively.The proposed method provides us with an intrinsic geometry viewpoint to understand the time series.     

Synchronization in networks of spatially extended systems

Synchronization processes in networks of spatially extended dynamical systems are analytically and numerically studied. We focus on the relevant case of networks whose elements (or nodes) are spatially extended dynamical systems, with the nodes being connected with each other by scalar signals. The stability of the synchronous spatio-temporal state for a generic network is analytically assessed by means of an extension of the master stability function approach. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. The efficiency and reliability of this method is illustrated numerically with networks of beam-plasma chaotic systems (Pierce diodes). We discuss also how the revealed regularities are expected to take place in other relevant physical and biological circumstances.     

Resurgence of oscillation in coupled oscillators under delayed cyclic interaction

This paper investigates the emergence of amplitude death and revival of oscillations from the suppression states in a system of coupled dynamical units interacting through delayed cyclic mode. In order to resurrect the oscillation from amplitude death state, we introduce asymmetry and feedback parameter in the cyclic coupling forms as a result of which the death region shrinks due to higher asymmetry and lower feedback parameter values for coupled oscillatory systems. Some analytical conditions are derived for amplitude death and revival of oscillations in two coupled limit cycle oscillators and corresponding numerical simulations confirm the obtained theoretical results. We also report that the death state and revival of oscillations from quenched state are possible in the network of identical coupled oscillators. The proposed mechanism has also been examined using chaotic Lorenz oscillator.     

Chaotic Eigenfunctions in Phase Space

We study individual eigenstates of quantized area-preserving maps on the 2-torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann–Husimi representations for quantum states as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellation of zeros of the Husimi density). We rigorously prove that a semiclassical uniform distribution of Husimi densities on the torus entails a similar equidistribution for the corresponding constellations. We deduce from this property a universal behavior for the phase patterns of chaotic Bargmann eigenfunctions which is reminiscent of the WKB approximation for eigenstates of integrable systems (though in a weaker sense). In order to obtain more precise information on chaotic eigenconstellations, we then model their properties by ensembles of random states, generalizing former results on the 2-sphere to the torus geometry. This approach yields statistical predictions for the constellations which fit quite well the chaotic data. We finally observe that specific dynamical information, e.g., the presence of high peaks (like scars) in Husimi densities, can be recovered from the knowledge of a few long-wavelength Fourier coefficients, which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions.     

Dynamics of Optical Bistability with Kerr-nonlinear Blackbody Radiation Reservoir

In this paper we investigate the nonlinear dynamics for optical bistabile(OB) model of homogeneously broadened two-level atomic medium interacting with a single mode of the ring cavity in the presence of a Kerr-nonlinear blackbody(KNB) radiation reservoir. We show the impact of the relative temperature of the reservoir on the transition between the dynamical states via bifurcation diagrams that represents the relation between maximum values of the output field and the relative temperature for fixed input field. Specifically, decreasing the relative temperature(T_b)causes the system to bifurcate from periodic to chaotic behavior and in turn reverts back to periodic behavior with further decrease of T_b. Varying atomic detuning leads to a change in the nature of the dynamic transition between the system's states from self pulsing to chaotic behavior.     

Dynamics of complex systems: scaling laws for the period of boolean networks

Boolean networks serve as models for complex systems, such as social or genetic networks, where each vertex, based on inputs received from selected vertices, makes its own decision about its state. Despite their simplicity, little is known about the dynamical properties of these systems. Here we propose a method to calculate the period of a finite Boolean system, by identifying the mechanisms determining its value. The proposed method can be applied to systems of arbitrary topology, and can serve as a roadmap for understanding the dynamics of large interacting systems in general.     

Synchronization of spatiotemporal chaos in a class of complex dynamical networks

This paper studies the synchronization of complex dynamical networks constructed by spatiotemporal chaotic systems with unknown parameters. The state variables in the systems with uncertain parameters are used to construct the parameter recognizers, and the unknown parameters are identified. Uncertain spatiotemporal chaotic systems are taken as the nodes of complex dynamical networks, connection among the nodes of all the spatiotemporal chaotic systems is of nonlinear coupling. The structure of the coupling functions between the connected nodes and the control gain are obtained based on Lyapunov stability theory. It is seen that stable chaos synchronization exists in the whole network when the control gain is in a certain range. The Gray--Scott models which have spatiotemporal chaotic behaviour are taken as examples for simulation and the results show that the method is very effective.     

Probing a subcritical instability with an amplitude expansion: An exploration of how far one can get

We explore methods to locate subcritical branches of spatially periodic solutions in pattern forming systems with a nonlinear finite-wavelength instability. We do so by means of a direct expansion in the amplitude of the linearly least stable mode about the appropriate reference state which one considers. This is motivated by the observation that for some equations fully nonlinear chaotic dynamics has been found to be organized around periodic solutions that do not simply bifurcate from the basic (laminar) state. We apply the method to two model equations, a subcritical generalization of the Swift–Hohenberg equation and a novel extension of the Kuramoto–Sivashinsky equation that we introduce to illustrate the abovementioned scenario in which weakly chaotic subcritical dynamics is organized around periodic states that bifurcate “from infinity” and that can nevertheless be probed perturbatively. We explore the reliability and robustness of such an expansion, with a particular focus on the use of these methods for determining the existence and approximate properties of finite-amplitude stationary solutions. Such methods obviously are to be used with caution: the expansions are often only asymptotic approximations, and if they converge their radius of convergence may be small. Nevertheless, expansions to higher order in the amplitude can be a useful tool to obtain qualitatively reliable results.     

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.