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A foundational machine-learning architecture is reinforcement learning, where an outstanding problem is achieving an optimal balance between exploration and exploitation. Specifically, exploration enables the agents to discover optimal policies in unknown domains of the environment for gaining potentially large future rewards, while exploitation relies on the already acquired knowledge to maximize the immediate rewards. We articulate an approach to this problem, treating the dynamical process of reinforcement learning as a Markov decision process that can be modeled as a nondeterministic finite automaton and defining a subset of states in the automaton to represent the preference for exploring unknown domains of the environment. Exploration is prioritized by assigning higher transition probabilities to these states. We derive a mathematical framework to systematically balance exploration and exploitation by formulating it as a mixed integer programming (MIP) problem to optimize the agent's actions and maximize the discovery of novel preferential states. Solving the MIP problem provides a trade-off point between exploiting known states and exploring unexplored regions. We validate the framework computationally with a benchmark system and argue that the articulated automaton is effectively an adaptive network with a time-varying connection matrix, where the states in the automaton are nodes and the transitions among the states represent the edges. The network is adaptive because the transition probabilities evolve over time. The established connection between the adaptive automaton arising from reinforcement learning and the adaptive network opens the door to applying theories of complex dynamical networks to address frontier problems in machine learning and artificial intelligence., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

A problem in nonlinear and complex dynamical systems with broad applications is forecasting the occurrence of a critical transition based solely on data without knowledge about the system equations. When such a transition leads to system collapse, as often is the case, all the available data are from the pre-critical regime where the system still functions normally, making the prediction problem challenging. In recent years, a machine-learning based approach tailored to solving this difficult prediction problem, adaptable reservoir computing, has been articulated. This Perspective introduces the basics of this machine-learning scheme and describes representative results. The general setting is that the system dynamics live on a normal attractor with oscillatory dynamics at the present time and, as a bifurcation parameter changes into the future, a critical transition can occur after which the system switches to a completely different attractor, signifying system collapse. To predict a critical transition, it is essential that the reservoir computer not only learns the dynamical "climate" of the system of interest at some specific parameter value but, more importantly, discovers how the system dynamics changes with the bifurcation parameter. It is demonstrated that this capability can be endowed into the machine through a training process with time series from a small number of distinct, pre-critical parameter values, thereby enabling accurate and reliable prediction of the catastrophic critical transition. Three applications are presented: predicting crisis, forecasting amplitude death, and creating digital twins of nonlinear dynamical systems. Limitations and future perspectives are discussed., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

Reconstructing complex networks and predicting the dynamics are particularly challenging in real-world applications because the available information and data are incomplete. We develop a unified collaborative deep-learning framework consisting of three modules: network inference, state estimation, and dynamical learning. The complete network structure is first inferred and the states of the unobserved nodes are estimated, based on which the dynamical learning module is activated to determine the dynamical evolution rules. An alternating parameter updating strategy is deployed to improve the inference and prediction accuracy. Our framework outperforms baseline methods for synthetic and empirical networks hosting a variety of dynamical processes. A reciprocity emerges between network inference and dynamical prediction: better inference of network structure improves the accuracy of dynamical prediction, and vice versa. We demonstrate the superior performance of our framework on an influenza dataset consisting of 37 US States and a PM2.5 dataset covering 184 cities in China., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

In the classic Kuramoto system of coupled two-dimensional rotators, chimera states characterized by the coexistence of synchronous and asynchronous groups of oscillators are long-lived because the average lifetime of these states increases exponentially with the system size. Recently, it was discovered that, when the rotators in the Kuramoto model are three-dimensional, the chimera states become short-lived in the sense that their lifetime scales with only the logarithm of the dimension-augmenting perturbation. We introduce transverse-stability analysis to understand the short-lived chimera states. In particular, on the unit sphere representing three-dimensional (3D) rotations, the long-lived chimera states in the classic Kuramoto system occur on the equator, to which latitudinal perturbations that make the rotations 3D are transverse. We demonstrate that the largest transverse Lyapunov exponent calculated with respect to these long-lived chimera states is typically positive, making them short-lived. The transverse-stability analysis turns the previous numerical scaling law of the transient lifetime into an exact formula: the "free" proportional constant in the original scaling law can now be precisely determined in terms of the largest transverse Lyapunov exponent. Our analysis reinforces the speculation that in physical systems, chimera states can be short-lived as they are vulnerable to any perturbations that have a component transverse to the invariant subspace in which they live., (© 2023 Author(s). Published under an exclusive license by AIP Publishing.)

We articulate the design imperatives for machine learning based digital twins for nonlinear dynamical systems, which can be used to monitor the "health" of the system and anticipate future collapse. The fundamental requirement for digital twins of nonlinear dynamical systems is dynamical evolution: the digital twin must be able to evolve its dynamical state at the present time to the next time step without further state input-a requirement that reservoir computing naturally meets. We conduct extensive tests using prototypical systems from optics, ecology, and climate, where the respective specific examples are a chaotic CO 2 laser system, a model of phytoplankton subject to seasonality, and the Lorenz-96 climate network. We demonstrate that, with a single or parallel reservoir computer, the digital twins are capable of a variety of challenging forecasting and monitoring tasks. Our digital twin has the following capabilities: (1) extrapolating the dynamics of the target system to predict how it may respond to a changing dynamical environment, e.g., a driving signal that it has never experienced before, (2) making continual forecasting and monitoring with sparse real-time updates under non-stationary external driving, (3) inferring hidden variables in the target system and accurately reproducing/predicting their dynamical evolution, (4) adapting to external driving of different waveform, and (5) extrapolating the global bifurcation behaviors to network systems of different sizes. These features make our digital twins appealing in applications, such as monitoring the health of critical systems and forecasting their potential collapse induced by environmental changes or perturbations. Such systems can be an infrastructure, an ecosystem, or a regional climate system.

Symmetries, due to their fundamental importance to dynamical processes on networks, have attracted a great deal of current research. Finding all symmetric nodes in large complex networks typically relies on automorphism groups from algebraic-group theory, which are solvable in quasipolynomial time. We articulate a conceptually appealing and computationally extremely efficient approach to finding and characterizing all symmetric nodes by introducing a structural position vector (SPV) for each node in networks. We establish the mathematical result that symmetric nodes must have the same SPV value and demonstrate, using six representative complex networks from the real world, that all symmetric nodes in these networks can be found in linear time. Furthermore, the SPVs not only characterize the similarity of nodes but also quantify the nodal influences in propagation dynamics. A caveat is that the proved mathematical result relating the SPV values to nodal symmetries is not sufficient; i.e., nodes having the same SPV values may not be symmetric, which arises in regular networks or networks with a dominant regular component. We point out with an analysis that this caveat is, in fact, shared by the known existing approaches to finding symmetric nodes in the literature. We further argue, with the aid of a mathematical analysis, that our SPV method is generally effective for finding the symmetric nodes in real-world networks that typically do not have a dominant regular component. Our SPV-based framework, therefore, provides a physically intuitive and computationally efficient way to uncover, understand, and exploit symmetric structures in complex networks arising from real-world applications.

Boolean networks introduced by Kauffman, originally intended as a prototypical model for gaining insights into gene regulatory dynamics, have become a paradigm for understanding a variety of complex systems described by binary state variables. However, there are situations, e.g., in biology, where a binary state description of the underlying dynamical system is inadequate. We propose random ternary networks and investigate the general dynamical properties associated with the ternary discretization of the variables. We find that the ternary dynamics can be either ordered or disordered with a positive Lyapunov exponent, and the boundary between them in the parameter space can be determined analytically. A dynamical event that is key to determining the boundary is the emergence of an additional fixed point for which we provide numerical verification. We also find that the nodes playing a pivotal role in shaping the system dynamics have characteristically distinct behaviors in different regions of the parameter space, and, remarkably, the boundary between these regions coincides with that separating the ordered and disordered dynamics. Overall, our framework of ternary networks significantly broadens the classical Boolean paradigm by enabling a quantitative description of richer and more complex dynamical behaviors.

We uncover a phenomenon in coupled nonlinear networks with a symmetry: as a bifurcation parameter changes through a critical value, synchronization among a subset of nodes can deteriorate abruptly, and, simultaneously, perfect synchronization emerges suddenly among a different subset of nodes that are not directly connected. This is a synchronization metamorphosis leading to an explosive transition to remote synchronization. The finding demonstrates that an explosive onset of synchrony and remote synchronization, two phenomena that have been studied separately, can arise in the same system due to symmetry, providing another proof that the interplay between nonlinear dynamics and symmetry can lead to a surprising phenomenon in physical systems.

Spatially distinct, self-sustained oscillations in artificial neural networks are fundamental to information encoding, storage, and processing in these systems. Here, we develop a method to induce a large variety of self-sustained oscillatory patterns in artificial neural networks and a controlling strategy to switch between different patterns. The basic principle is that, given a complex network, one can find a set of nodes-the minimum feedback vertex set (mFVS), whose removal or inhibition will result in a tree-like network without any loop structure. Reintroducing a few or even a single mFVS node into the tree-like artificial neural network can recover one or a few of the loops and lead to self-sustained oscillation patterns based on these loops. Reactivating various mFVS nodes or their combinations can then generate a large number of distinct neuronal firing patterns with a broad distribution of the oscillation period. When the system is near a critical state, chaos can arise, providing a natural platform for pattern switching with remarkable flexibility. With mFVS guided control, complex networks of artificial neurons can thus be exploited as potential prototypes for local, analog type of processing paradigms.

In applications of nonlinear and complex dynamical systems, a common situation is that the system can be measured, but its structure and the detailed rules of dynamical evolution are unknown. The inverse problem is to determine the system equations and structure from time series. The principle of exploiting sparse optimization to find the equations of dynamical systems from data was first articulated in 2011 by the ASU group. The basic idea is to expand the system equations into a power series or a Fourier series of a finite number of terms and then to determine the vector of the expansion coefficients based solely on data through sparse optimization. This Tutorial presents a brief review of the recent progress in this area. Issues discussed include discovering the equations of stationary or nonstationary chaotic systems to enable the prediction of critical transition and system collapse, inferring the full topology of complex oscillator networks and social networks hosting evolutionary game dynamics, and identifying partial differential equations for spatiotemporal dynamical systems. Situations where sparse optimization works or fails are pointed out. The relation with the traditional delay-coordinate embedding method is discussed, and the recent development of a model-free, data-driven prediction framework based on machine learning is mentioned.

Recent interest in exploiting machine learning for model-free prediction of chaotic systems focused on the time evolution of the dynamical variables of the system as a whole, which include both amplitude and phase. In particular, in the framework based on reservoir computing, the prediction horizon as determined by the largest Lyapunov exponent is often short, typically about five or six Lyapunov times that contain approximately equal number of oscillation cycles of the system. There are situations in the real world where the phase information is important, such as the ups and downs of species populations in ecology, the polarity of a voltage variable in an electronic circuit, and the concentration of certain chemical above or below the average. Using classic chaotic oscillators and a chaotic food-web system from ecology as examples, we demonstrate that reservoir computing can be exploited for long-term prediction of the phase of chaotic oscillators. The typical prediction horizon can be orders of magnitude longer than that with predicting the entire variable, for which we provide a physical understanding. We also demonstrate that a properly designed reservoir computing machine can reliably sense phase synchronization between a pair of coupled chaotic oscillators with implications to the design of the parallel reservoir scheme for predicting large chaotic systems.

It is common knowledge that alcohol consumption during pregnancy would cause cognitive impairment in children. However, recent works suggested that the risk of drinking during pregnancy may have been exaggerated. It is critical to determine whether and up to which amount the consumption of alcohol will affect the cognitive development of children. We evaluate time-varying functional connectivity using magnetoencephalogram data from somatosensory evoked response experiments for 19 teenage subjects with prenatal alcohol exposure and 21 healthy control teenage subjects using a new time-varying connectivity approach, combining renormalised partial directed coherence with state space modeling. Children exposed to alcohol prenatally are at risk of developing a Fetal Alcohol Spectrum Disorder (FASD) characterized by cerebral connectivity deficiency and impaired cognitive abilities. Through a comparison study of teenage subjects exposed to alcohol prenatally with healthy control subjects, we establish that the inter-hemispheric connectivity is deficient for the former, which may lead to disruption in the cortical inter-hemispheric connectivity and deficits in higher order cognitive functions as measured by an IQ test, for example. We provide quantitative evidence that the disruption is correlated with cognitive deficits. These findings could lead to a novel, highly sensitive biomarker for FASD and support a recommendation of no safe amount of alcohol consumption during pregnancy.

We analyze five big data sets from a variety of online social networking (OSN) systems and find that the growth dynamics of meme popularity exhibit characteristically different behaviors. For example, there is linear growth associated with online recommendation and sharing platforms, a plateaued (or an "S"-shape) type of growth behavior in a web service devoted to helping users to collect bookmarks, and an exponential increase on the largest and most popular microblogging website in China. Does a universal mechanism with a common set of dynamical rules exist, which can explain these empirically observed, distinct growth behaviors? We provide an affirmative answer in this paper. In particular, inspired by biomimicry to take advantage of cell population growth dynamics in microbial ecology, we construct a base growth model for meme popularity in OSNs. We then take into account human factors by incorporating a general model of human interest dynamics into the base model. The final hybrid model contains a small number of free parameters that can be estimated purely from data. We demonstrate that our model is universal in the sense that, with a few parameters estimated from data, it can successfully predict the distinct meme growth dynamics. Our study represents a successful effort to exploit principles in biology to understand online social behaviors by incorporating the traditional microbial growth model into meme popularity. Our model can be used to gain insights into critical issues such as classification, robustness, optimization, and control of OSN systems.

Quantum chaos is referred to as the study of quantum manifestations or fingerprints of classical chaos. A vast majority of the studies were for nonrelativistic quantum systems described by the Schrödinger equation. Recent years have witnessed a rapid development of Dirac materials such as graphene and topological insulators, which are described by the Dirac equation in relativistic quantum mechanics. A new field has thus emerged: relativistic quantum chaos. This Tutorial aims to introduce this field to the scientific community. Topics covered include scarring, chaotic scattering and transport, chaos regularized resonant tunneling, superpersistent currents, and energy level statistics-all in the relativistic quantum regime. As Dirac materials have the potential to revolutionize solid-state electronic and spintronic devices, a good understanding of the interplay between chaos and relativistic quantum mechanics may lead to novel design principles and methodologies to enhance device performance.

One of the most challenging problems in network science is to accurately detect communities at distinct hierarchical scales. Most existing methods are based on structural analysis and manipulation, which are NP-hard. We articulate an alternative, dynamical evolution-based approach to the problem. The basic principle is to computationally implement a nonlinear dynamical process on all nodes in the network with a general coupling scheme, creating a networked dynamical system. Under a proper system setting and with an adjustable control parameter, the community structure of the network would "come out" or emerge naturally from the dynamical evolution of the system. As the control parameter is systematically varied, the community hierarchies at different scales can be revealed. As a concrete example of this general principle, we exploit clustered synchronization as a dynamical mechanism through which the hierarchical community structure can be uncovered. In particular, for quite arbitrary choices of the nonlinear nodal dynamics and coupling scheme, decreasing the coupling parameter from the global synchronization regime, in which the dynamical states of all nodes are perfectly synchronized, can lead to a weaker type of synchronization organized as clusters. We demonstrate the existence of optimal choices of the coupling parameter for which the synchronization clusters encode accurate information about the hierarchical community structure of the network. We test and validate our method using a standard class of benchmark modular networks with two distinct hierarchies of communities and a number of empirical networks arising from the real world. Our method is computationally extremely efficient, eliminating completely the NP-hard difficulty associated with previous methods. The basic principle of exploiting dynamical evolution to uncover hidden community organizations at different scales represents a "game-change" type of approach to addressing the problem of community detection in complex networks.

We present a novel class of nonlinear dynamical systems-a hybrid of relativistic quantum and classical systems and demonstrate that multistability is ubiquitous. A representative setting is coupled systems of a topological insulator and an insulating ferromagnet, where the former possesses an insulating bulk with topologically protected, dissipationless, and conducting surface electronic states governed by the relativistic quantum Dirac Hamiltonian and the latter is described by the nonlinear classical evolution of its magnetization vector. The interactions between the two are essentially the spin transfer torque from the topological insulator to the ferromagnet and the local proximity induced exchange coupling in the opposite direction. The hybrid system exhibits a rich variety of nonlinear dynamical phenomena besides multistability such as bifurcations, chaos, and phase synchronization. The degree of multistability can be controlled by an external voltage. In the case of two coexisting states, the system is effectively binary, opening a door to exploitation for developing spintronic memory devices. Because of the dissipationless and spin-momentum locking nature of the surface currents of the topological insulator, little power is needed for generating a significant current, making the system appealing for potential applications in next generation of low power memory devices.

Switching dynamics among saddles in a network of nonlinear oscillators can be exploited for information encoding and processing (hence computing), but stable attractors in the system can terminate the switching behavior. An effective control strategy is presented to sustain switching dynamics in networks of pulse-coupled oscillators. The support for the switching behavior is a set of saddles, or unstable invariant sets in the phase space. We thus identify saddles with a common property, localize the system in the vicinity of them, and then guide the system from one metastable state to another to generate desired switching dynamics. We demonstrate that the control method successfully generates persistent switching trajectories and prevents the system from entering stable attractors. In addition, there exists correspondence between the network structure and the switching dynamics, providing fundamental insights on the development of a computing paradigm based on the switching dynamics.

The interplay between individual behaviors and epidemic dynamics in complex networks is a topic of recent interest. In particular, individuals can obtain different types of information about the disease and respond by altering their behaviors, and this can affect the spreading dynamics, possibly in a significant way. We propose a model where individuals' behavioral response is based on a generic type of local information, i.e., the number of neighbors that has been infected with the disease. Mathematically, the response can be characterized by a reduction in the transmission rate by a factor that depends on the number of infected neighbors. Utilizing the standard susceptible-infected-susceptible and susceptible-infected-recovery dynamical models for epidemic spreading, we derive a theoretical formula for the epidemic threshold and provide numerical verification. Our analysis lays on a solid quantitative footing the intuition that individual behavioral response can in general suppress epidemic spreading. Furthermore, we find that the hub nodes play the role of "double-edged sword" in that they can either suppress or promote outbreak, depending on their responses to the epidemic, providing additional support for the idea that these nodes are key to controlling epidemic spreading in complex networks.

A fundamental result in the evolutionary-game paradigm of cyclic competition in spatially extended ecological systems, as represented by the classic Reichenbach-Mobilia-Frey (RMF) model, is that high mobility tends to hamper or even exclude species coexistence. This result was obtained under the hypothesis that individuals move randomly without taking into account the suitability of their local environment. We incorporate local habitat suitability into the RMF model and investigate its effect on coexistence. In particular, we hypothesize the use of "basic instinct" of an individual to determine its movement at any time step. That is, an individual is more likely to move when the local habitat becomes hostile and is no longer favorable for survival and growth. We show that, when such local habitat suitability is taken into account, robust coexistence can emerge even in the high-mobility regime where extinction is certain in the RMF model. A surprising finding is that coexistence is accompanied by the occurrence of substantial empty space in the system. Reexamination of the RMF model confirms the necessity and the important role of empty space in coexistence. Our study implies that adaptation/movements according to local habitat suitability are a fundamental factor to promote species coexistence and, consequently, biodiversity.

Chaos has long been recognized to be generally advantageous from the perspective of control. In particular, the infinite number of unstable periodic orbits embedded in a chaotic set and the intrinsically sensitive dependence on initial conditions imply that a chaotic system can be controlled to a desirable state by using small perturbations. Investigation of chaos control, however, was largely limited to nonlinear dynamical systems in the classical realm. In this paper, we show that chaos may be used to modulate or harness quantum mechanical systems. To be concrete, we focus on quantum transport through nanostructures, a problem of considerable interest in nanoscience, where a key feature is conductance fluctuations. We articulate and demonstrate that chaos, more specifically transient chaos, can be effective in modulating the conductance-fluctuation patterns. Experimentally, this can be achieved by applying an external gate voltage in a device of suitable geometry to generate classically inaccessible potential barriers. Adjusting the gate voltage allows the characteristics of the dynamical invariant set responsible for transient chaos to be varied in a desirable manner which, in turn, can induce continuous changes in the statistical characteristics of the quantum conductance-fluctuation pattern. To understand the physical mechanism of our scheme, we develop a theory based on analyzing the spectrum of the generalized non-Hermitian Hamiltonian that includes the effect of leads, or electronic waveguides, as self-energy terms. As the escape rate of the underlying non-attracting chaotic set is increased, the imaginary part of the complex eigenenergy becomes increasingly large so that pointer states are more difficult to form, making smoother the conductance-fluctuation pattern.

Branched wave structures, an unconventional wave propagation pattern, can arise in random media. Experimental evidence has accumulated, revealing the occurrence of these waves in systems ranging from microwave and optical systems to solid-state devices. Experiments have also established the universal feature that the wave-intensity statistics deviate from Gaussian and typically possess a long-tail distribution, implying the existence of spatially localized regions with extraordinarily high intensity concentration ("hot" spots). Despite previous efforts, the origin of branched wave pattern is currently an issue of debate. Recently, we proposed a "minimal" model of wave propagation and scattering in optical media, taking into account the essential physics for generating robust branched flows: (1) a finite-size medium for linear wave propagation and (2) random scatterers whose refractive indices deviate continuously from that of the background medium. Here we provide extensive numerical evidence and a comprehensive analytic treatment of the scaling behavior to establish that branched wave patterns can emerge as a general phenomenon in wide parameter regime in between the weak-scattering limit and Anderson localization. The basic physical mechanisms to form branched waves are breakup of waves by a single scatterer and constructive interference of broken waves from multiple scatterers. Despite simplicity of our model, analysis of the scattering field naturally yields an algebraic (power-law) statistic in the high wave-intensity distribution, indicating that our model is able to capture the generic physical origin of these special wave patterns. The insights so obtained can be used to better understand the origin of complex extreme wave patterns, whose occurrences can have significant impact on the performance of the underlying physical systems or devices.

Recent studies have suggested the necessity to incorporate traffic dynamics into the process of epidemic spreading on complex networks, as the former provides support for the latter in many real-world situations. While there are results on the asymptotic scope of the spreading dynamics, the issue of how fast an epidemic outbreak can occur remains outstanding. We observe numerically that the density of the infected nodes exhibits an exponential increase with time initially, rendering definable a characteristic time for the outbreak. We then derive a formula for scale-free networks, which relates this time to parameters characterizing the traffic dynamics and the network structure such as packet-generation rate and betweenness distribution. The validity of the formula is tested numerically. Our study indicates that increasing the average degree and/or inducing traffic congestion can slow down the spreading process significantly.

Nonlinear dynamical systems in reality are often under environmental influences that are time-dependent. To assess whether such a system can perform as desired or as designed and is sustainable requires forecasting its future states and attractors based solely on time series. We propose a viable solution to this challenging problem by resorting to the compressive-sensing paradigm. In particular, we demonstrate that, for a dynamical system whose equations are unknown, a series expansion in both dynamical and time variables allows the forecasting problem to be formulated and solved in the framework of compressive sensing using only a few measurements. We expect our method to be useful in addressing issues of significant current concern such as the sustainability of various natural and man-made systems.

Reverse engineering of complex dynamical networks is important for a variety of fields where uncovering the full topology of unknown networks and estimating parameters characterizing the network structure and dynamical processes are of interest. We consider complex oscillator networks with time-delayed interactions in a noisy environment, and develop an effective method to infer the full topology of the network and evaluate the amount of time delay based solely on noise-contaminated time series. In particular, we develop an analytic theory establishing that the dynamical correlation matrix, which can be constructed purely from time series, can be manipulated to yield both the network topology and the amount of time delay simultaneously. Extensive numerical support is provided to validate the method. While our method provides a viable solution to the network inverse problem, significant difficulties, limitations, and challenges still remain, and these are discussed thoroughly.

We study catastrophic behaviors in large networked systems in the paradigm of evolutionary games by incorporating a realistic "death" or "bankruptcy" mechanism. We find that a cascading bankruptcy process can arise when defection strategies exist and individuals are vulnerable to deficit. Strikingly, we observe that, after the catastrophic cascading process terminates, cooperators are the sole survivors, regardless of the game types and of the connection patterns among individuals as determined by the topology of the underlying network. It is necessary that individuals cooperate with each other to survive the catastrophic failures. Cooperation thus becomes the optimal strategy and absolutely outperforms defection in the game evolution with respect to the "death" mechanism. Our results can be useful for understanding large-scale catastrophe in real-world systems and in particular, they may yield insights into significant social and economical phenomena such as large-scale failures of financial institutions and corporations during an economic recession.

We present a general method to analyze multichannel time series that are becoming increasingly common in many areas of science and engineering. Of particular interest is the degree of synchrony among various channels, motivated by the recognition that characterization of synchrony in a system consisting of many interacting components can provide insights into its fundamental dynamics. Often such a system is complex, high-dimensional, nonlinear, nonstationary, and noisy, rendering unlikely complete synchronization in which the dynamical variables from individual components approach each other asymptotically. Nonetheless, a weaker type of synchrony that lasts for a finite amount of time, namely, phase synchronization, can be expected. Our idea is to calculate the average phase-synchronization times from all available pairs of channels and then to construct a matrix. Due to nonlinearity and stochasticity, the matrix is effectively random. Moreover, since the diagonal elements of the matrix can be arbitrarily large, the matrix can be singular. To overcome this difficulty, we develop a random-matrix based criterion for proper choosing of the diagonal matrix elements. Monitoring of the eigenvalues and the determinant provides a powerful way to assess changes in synchrony. The method is tested using a prototype nonstationary noisy dynamical system, electroencephalogram (scalp) data from absence seizures for which enhanced cortico-thalamic synchrony is presumed, and electrocorticogram (intracranial) data from subjects having partial seizures with secondary generalization for which enhanced local synchrony is similarly presumed.

We investigate the dynamics of light rays in two classes of optical metamaterial systems: (1) time-dependent system with a volcano-shaped, inhomogeneous and isotropic refractive-index distribution, subject to external electromagnetic perturbations and (2) time-independent system consisting of three overlapping or non-overlapping refractive-index distributions. Utilizing a mechanical-optical analogy and coordinate transformation, the wave-propagation problem governed by the Maxwell's equations can be modeled by a set of ordinary differential equations for light rays. We find that transient chaotic dynamics, hyperbolic or nonhyperbolic, are common in optical metamaterial systems. Due to the analogy between light-ray dynamics in metamaterials and the motion of light in matter as described by general relativity, our results reinforce the recent idea that chaos in gravitational systems can be observed and studied in laboratory experiments.

Recently, small-scale quantum communication has been realized by transporting entangled photons, rendering potentially feasible quantum networks on large scales. We propose a class of quantum networks comprising quantum repeaters for long-distance information transport and local networks of fibers and switches. As the transmitting capability is limited by the node bandwidth, photon loss can occur through the dynamical process of cascading when the network is under external perturbations. We obtain results that can be used to guide quantum network design to minimize the photon loss.

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions.

Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system., (© 2010 American Institute of Physics.)

We propose a scheme to induce intrinsic localized modes (ILMs) at an arbitrary site in microelectromechanical cantilever arrays. The idea is to locate the particular cantilever beam in the array that one wishes to drive to an oscillating state with significantly higher amplitude than the average and then apply small adjustments to the electrical signal that drives the whole array system. Our scheme is thus a global closed-loop control strategy. We argue that the dynamical mechanism on which our global driving scheme relies is spatiotemporal chaos and we develop a detailed analysis based on the standard averaging method in nonlinear dynamics to understand the working of our control scheme. We also develop a Markov model to characterize the transient time required for inducing ILMs., (© 2010 American Institute of Physics.)

Evolutionary-game based models of nonhierarchical, cyclically competing populations have become paradigmatic for addressing the fundamental problem of species coexistence in spatially extended ecosystems. We study the role of intraspecific competition in the coexistence and find that the competition can strongly promote the coexistence for high individual mobility in the sense that stable coexistence can arise in parameter regime where extinction would occur without the competition. The critical value of the competition rate beyond which the coexistence is induced is found to be independent of the mobility. We derive a theoretical model based on nonlinear partial differential equations to predict the critical competition rate and the boundaries between the coexistence and extinction regions in a relevant parameter space. We also investigate pattern formation and well-mixed spatiotemporal population dynamics to gain further insights into our findings., ((c) 2010 American Institute of Physics.)

Previous works on traffic-flow dynamics on complex networks have mostly focused on continuous phase transition from a free-flow state to a locally congested state as a parameter, such as the packet-generating rate, is increased through a critical value. Above the transition point congestion occurs on a small subset of nodes. Utilizing a conventional traffic-flow model based on the packet birth-death process and more importantly, taking into account the fact that in realistic networks nodes have only finite buffers, we find an abrupt transition from free flow to complete congestion. Slightly below the transition point, the network can support the maximum amount of traffic for some optimal value of the routing parameter. We develop a mean-field theory to explain the surprising transition phenomenon and provide numerical support. Furthermore, we propose a control strategy based on the idea of random packet dropping to prevent/break complete congestion. Our finding provides insights into realistic communication networks where complete congestion can occur directly from a free-flow state without any apparent precursor, and our control strategy can be effective to restore traffic flow once complete congestion has occurred.

Experimental evidence of intrinsic localized modes (ILMs) in microelectromechanical oscillator arrays has been reported recently. In this paper, we carry out a detailed analysis of a new mechanism for ILMs in typical experimental settings; that is, spatiotemporal chaos is ubiquitous and it provides a natural platform for actual realization of various ILMs through frequency control. We find that unstable periodic orbits associated with ILMs are pivotal for spatiotemporal chaos to arise and these orbits are the keys to stabilizing ILMs by frequency modulation.

We investigate Kuramoto dynamics on scale-free networks to include the effect of weights, as weighted networks are conceivably more pertinent to real-world situations than unweighted networks. We consider both symmetric and asymmetric coupling schemes. Our analysis and computations indicate that more links in weighted scale-free networks can either promote or suppress synchronization. In particular, we find that as a parameter characterizing the weighting scheme is varied, there can be two distinct regimes: a normal regime where more links can enhance synchronization and an abnormal regime where the opposite occurs. A striking phenomenon is that for dense networks for which the mean-field approximation is satisfied, the point separating the two regimes does not depend on the details of the network structure such as the average degree and the degree exponent. This implies the existence of a class of weighted scale-free networks for which the synchronization dynamics are invariant with respect to the network properties. We also perform a comparison study with respect to the onset of synchronization in Kuramoto networks and the synchronization stability of networks of identical oscillators.

Multistability has been a phenomenon of continuous interest in nonlinear dynamics. Most existing works so far have focused on smooth dynamical systems. Motivated by the fact that nonsmooth dynamical systems can arise commonly in realistic physical and engineering applications such as impact oscillators and switching electronic circuits, we investigate multistability in such systems. In particular, we consider a generic class of piecewise smooth dynamical systems expressed in normal form but representative of nonsmooth systems in realistic situations, and focus on the weakly dissipative regime and the Hamiltonian limit. We find that, as the Hamiltonian limit is approached, periodic attractors can be generated through a series of saddle-node bifurcations. A striking phenomenon is that the periods of the newly created attractors follow an arithmetic sequence. This has no counterpart in smooth dynamical systems. We provide physical analyses, numerical computations, and rigorous mathematical arguments to substantiate the finding.

Complex clustered networks arise in biological, social, physical, and technological systems, and the synchronous dynamics on such networks have attracted recent interests. Here we investigate system-size dependence of the synchronizability of these networks. Theoretical analysis and numerical computations reveal that, for a typical clustered network, as its size is increased, the synchronizability can be maintained or even enhanced but at the expense of deterioration of the clustered characteristics in the topology that distinguish this type of networks from other types of complex networks. An implication is that, for a large network in a realistic situation, if synchronization is important for its function, then most likely it will not have a clustered topology.

Recently, it has been found that the synchronizability of a scale-free network can be enhanced by introducing some proper gradient in the coupling. This result has been obtained by using eigenvalue-spectrum analysis under the assumption of identical node dynamics. Here we obtain an analytic formula for the onset of synchronization by incorporating the Kuramoto model on gradient scale-free networks. Our result provides quantitative support for the enhancement of synchronization in such networks, further justifying their ubiquity in natural and in technological systems., ((c) 2008 American Institute of Physics.)

Detecting a weak signal from chaotic time series is of general interest in science and engineering. In this work we introduce and investigate a signal detection algorithm for which chaos theory, nonlinear dynamical reconstruction techniques, neural networks, and time-frequency analysis are put together in a synergistic manner. By applying the scheme to numerical simulation and different experimental measurement data sets (Henon map, chaotic circuit, and NH(3) laser data sets), we demonstrate that weak signals hidden beneath the noise floor can be detected by using a model-based detector. Particularly, the signal frequencies can be extracted accurately in the time-frequency space. By comparing the model-based method with the standard denoising wavelet technique as well as supervised principal components analysis detector, we further show that the nonlinear dynamics and neural network-based approach performs better in extracting frequencies of weak signals hidden in chaotic time series.

There has been mounting evidence that many types of biological or technological networks possess a clustered structure. As many system functions depend on synchronization, it is important to investigate the synchronizability of complex clustered networks. Here we focus on one fundamental question: Under what condition can the network synchronizability be optimized? In particular, since the two basic parameters characterizing a complex clustered network are the probabilities of intercluster and intracluster connections, we investigate, in the corresponding two-dimensional parameter plane, regions where the network can be best synchronized. Our study yields a quite surprising finding: a complex clustered network is most synchronizable when the two probabilities match each other approximately. Mismatch, for instance caused by an overwhelming increase in the number of intracluster links, can counterintuitively suppress or even destroy synchronization, even though such an increase tends to reduce the average network distance. This phenomenon provides possible principles for optimal synchronization on complex clustered networks. We provide extensive numerical evidence and an analytic theory to establish the generality of this phenomenon.

For a microelectromechanical (MEM) resonator, the combination of mechanical nonlinearity and electrical driving force can lead to bistability. In such a case, the system exhibits two coexisting stable oscillatory states (attractors): one with low and another with high energy. Under the influence of noise, with high probability the system can be perturbed into the low-energy state. We propose a robust control scheme to place the system in the high-energy state. Our idea is not to pull the system out of the bistable regime but instead to take advantage of the nonlinear dynamics to achieve high-energy output. In particular, our control scheme consists of two steps: bifurcation control that temporarily drives the system to a regime with only one attractor, one that is the continuation of the high-energy attractor in the bistable regime; and ramping parameter control that restores the bistability while maintaining the system in the high-energy attractor. We derive an analytic theory to guide the control, provide numerical examples, and suggest a practical method to realize the control experimentally. Our result may find potential usage in devices based on MEM resonators where high output energy is desired.

To design complex networks to minimize traffic congestion, it is necessary to understand how traffic flow depends on network structure. We study data packet flow on complex networks, where the packet delivery capacity of each node is not fixed. The optimal configuration of capacities to minimize traffic congestion is derived and the critical packet generating rate is determined, below which the network is at a free flow state but above which congestion occurs. Our analysis reveals a direct relation between network topology and traffic flow. Optimal network structure, free of traffic congestion, should have two features: uniform distribution of load over all nodes and small network diameter. This finding is confirmed by numerical simulations. Our analysis also makes it possible to theoretically compare the congestion conditions for different types of complex networks. In particular, we find that network with low critical generating rate is more susceptible to congestion. The comparison has been made on the following complex-network topologies: random, scale-free, and regular.

It has been found recently that the averaged phase-synchronization time between the input and the output signals of a nonlinear dynamical system can exhibit an extremely high sensitivity to variations in the noise level. In real-world signal-processing applications, sensitivity to frequency variations may be of considerable interest. Here we investigate the dependence of the averaged phase-synchronization time on frequency of the input signal. Our finding is that, for typical nonlinear oscillator systems, there can be a frequency regime where the time exhibits significant sensitivity to frequency variations. We obtain an analytic formula to quantify the frequency dependence, provide numerical support, and present experimental evidence from a simple nonlinear circuit system.

Blind spots in sensor networks, i.e., individual nodes or small groups of nodes isolated from the rest of the network, are of great concern as they may significantly degrade the network's ability to collect and process information. As the operations of many types of sensors in realistic applications rely on short-lifetime power supplies (e.g., batteries), once they are used up ("off"), replacements become necessary ("on"). This off-and-on process can lead to blind spots. An issue of both theoretical and practical interest concerns the dynamical process and the critical behavior associated with the occurrence of blind spots. In particular, there can be various network topologies, and the off-and-on process can be characterized by the probability that a node functions normally, or the occupying probability of a node in the network. The question to be addressed in this paper concerns how the dynamics of blind spots depend on the network topology and on the occupying probability. For regular, random, and mixed networks, we provide theoretical formulas relating the probability of blind spots to the occupying probability, from which the critical point for the occurrence of blind spots can be determined. For scale-free networks, we present a procedure to estimate the critical point. While our theoretical and numerical analyses are presented in the framework of sensor networks, we expect our results to be generally applicable to network partitioning issues in other networks, such as the wireless cellular network, the Internet, or transportation networks, where the issue of blind spots may be of concern.

We report a general phenomenon concerning the effect of noise on phase synchronization in coupled chaotic oscillators: the average phase-synchronization time exhibits a nonmonotonic behavior with the noise amplitude. In particular, we find that the time exhibits a local minimum for relatively small noise amplitude but a local maximum for stronger noise. We provide numerical results, experimental evidence from coupled chaotic circuits, and a heuristic argument to establish the generality of this phenomenon.

It is believed that both Hebbian and homeostatic mechanisms are essential in neural learning. While Hebbian plasticity selectively modifies synaptic connectivity according to activity experienced, homeostatic plasticity constrains this change so that neural activity is always within reasonable physiological limits. Recent experiments reveal spike timing-dependent plasticity (STDP) as a new type of Hebbian learning with high time precision and heterosynaptic plasticity (HSP) as a new homeostatic mechanism acting directly on synapses. Here, we study the effect of STDP and HSP on randomly connected neural networks. Despite the reported successes of STDP to account for neural activities at the single-cell level, we find that, surprisingly, at the network level, networks trained using STDP alone cannot seem to generate realistic neural activities. For instance, STDP would stipulate that past sensory experience be maintained forever if it is no longer activated. To overcome this difficulty, motivated by the fact that HSP can induce strong competition between sensory experiences, we propose a biophysically plausible learning rule by combining STDP and HSP. Based on the Fokker-Planck theory and extensive numerical computations, we demonstrate that HSP and STDP operated on different time scales can complement each other, resulting in more realistic network activities. Our finding may provide fresh insight into the learning mechanism of the brain.

Chaotic scattering in open Hamiltonian systems under weak dissipation is not only of fundamental interest but also important for problems of current concern such as the advection and transport of inertial particles in fluid flows. Previous work using discrete maps demonstrated that nonhyperbolic chaotic scattering is structurally unstable in the sense that the algebraic decay of scattering particles immediately becomes exponential in the presence of weak dissipation. Here we extend the result to continuous-time Hamiltonian systems by using the Henon-Heiles system as a prototype model. More importantly, we go beyond to investigate the basin structure of scattering dynamics. A surprising finding is that, in the common case where multiple destinations exist for scattering trajectories, Wada basin boundaries are common and they appear to be structurally stable under weak dissipation, even when other characteristics of the nonhyperbolic scattering dynamics are not. We provide numerical evidence and a geometric theory for the structural stability of the complex basin topology.

Resonant perturbations are effective for harnessing nonlinear oscillators for various applications such as controlling chaos and inducing chaos. Of physical interest is the effect of small frequency mismatch on the attractors of the underlying dynamical systems. By utilizing a prototype of nonlinear oscillators, the periodically forced Duffing oscillator and its variant, we find a phenomenon: resonant-frequency mismatch can result in attractors that are nonchaotic but are apparently strange in the sense that they possess a negative Lyapunov exponent but its information dimension measured using finite numerics assumes a fractional value. We call such attractors pseudo-strange. The transition to pesudo-strange attractors as a system parameter changes can be understood analytically by regarding the system as nonstationary and using the Melnikov function. Our results imply that pseudo-strange attractors are common in nonstationary dynamical systems.

Networks with a community (or modular) structure arise in social and biological sciences. In such a network individuals tend to form local communities, each having dense internal connections. The linkage among the communities is, however, much more sparse. The dynamics on modular networks, for instance synchronization, may be of great social or biological interest. (Here by synchronization we mean some synchronous behavior among the nodes in the network, not, for example, partially synchronous behavior in the network or the synchronizability of the network with some external dynamics.) By using a recent theoretical framework, the master-stability approach originally introduced by Pecora and Carroll in the context of synchronization in coupled nonlinear oscillators, we address synchronization in complex modular networks. We use a prototype model and develop scaling relations for the network synchronizability with respect to variations of some key network structural parameters. Our results indicate that random, long-range links among distant modules is the key to synchronization. As an application we suggest a viable strategy to achieve synchronous behavior in social networks.