Emergent order in adaptively rewired networks.

We explore adaptive link change strategies that can lead a system to network configurations that yield ordered dynamical states. We propose two adaptive strategies based on feedback from the global synchronization error. In the first strategy, the connectivity matrix changes if the instantaneous synchronization error is larger than a prescribed threshold. In the second strategy, the probability of a link changing at any instant of time is proportional to the magnitude of the instantaneous synchronization error. We demonstrate that both these strategies are capable of guiding networks to chaos suppression within a prescribed tolerance, in two prototypical systems of coupled chaotic maps. So, the adaptation works effectively as an efficient search in the vast space of connectivities for a configuration that serves to yield a targeted pattern. The mean synchronization error shows the presence of a sharply defined transition to very low values after a critical coupling strength, in all cases. For the first strategy, the total time during which a network undergoes link adaptation also exhibits a distinct transition to a small value under increasing coupling strength. Analogously, for the second strategy, the mean fraction of links that change in the network over time, after transience, drops to nearly zero, after a critical coupling strength, implying that the network reaches a static link configuration that yields the desired dynamics. These ideas can then potentially help us to devise control methods for extended interactive systems, as well as suggest natural mechanisms capable of regularizing complex networks.

Robustness of the emergence of synchronized clusters in branching hierarchical systems under parametric noise.

The dynamical robustness of networks in the presence of noise is of utmost fundamental and applied interest. In this work, we explore the effect of parametric noise on the emergence of synchronized clusters in diffusively coupled Chaté-Manneville maps on a branching hierarchical structure. We consider both quenched and dynamically varying parametric noise. We find that the transition to a synchronized fixed point on the maximal cluster is robust in the presence of both types of noise. We see that the small sub-maximal clusters of the system, which coexist with the maximal cluster, exhibit a power-law cluster size distribution. This power-law scaling of synchronized cluster sizes is robust against noise in a broad range of coupling strengths. However, interestingly, we find a window of coupling strength where the system displays markedly different sensitivities to noise for the maximal cluster and the small clusters, with the scaling exponent for the cluster distribution for small clusters exhibiting clear dependence on noise strength, while the cluster size of the maximal cluster of the system displays no significant change in the presence of noise. Our results have implications for the observability of synchronized cluster distributions in real-world hierarchical networks, such as neural networks, power grids, and communication networks, that necessarily have parametric fluctuations.

Impact of coupling on neuronal extreme events: Mitigation and enhancement.

We focus on the emergence of extreme events in a collection of aperiodic neuronal maps, under local diffusive coupling, as well as global mean-field coupling. Our central finding is that local diffusive coupling enhances the probability of occurrence of both temporal and spatial extreme events, while in marked contrast, global mean-field coupling suppresses extreme events. So the nature of the coupling crucially determines whether the extreme events are enhanced or mitigated by coupling. Further, in globally coupled systems, there exist initial states in a window of coupling strength that exhibit spatial extreme events, but not temporal extreme events, suggesting that spatial extreme events do not imply temporal extreme events. We also explored the existence of discernible patterns in the return maps of successive inter-event intervals in order to gauge short-term risk-assessment. We find that single neuronal maps, as well as systems under strong diffusive coupling, display broad noisy patterns in these return maps, with clusters around characteristic intervals, allowing some short-term predictability in the extreme event sequence. In contrast, under weak diffusive coupling and global coupling, inter-event intervals lose all perceptible correlations, and the distribution extends to very large inter-event intervals. Lastly, we investigated a non-local diffusive coupling form. Interestingly, this coupling yielded a large window where temporal extreme events occurred, but the spatial profile was synchronized, namely, we found synchronized temporal extreme events. Such synchronized extreme spiking is reminiscent of the neuronal activity leading to epileptic seizures and is of potential relevance to extreme events in brain activity.

Construction of logic gates exploiting resonance phenomena in nonlinear systems.

A two-state system driven by two inputs has been found to consistently produce a response mirroring a logic function of the two inputs, in an optimal window of moderate noise. This phenomenon is called logical stochastic resonance (LSR). We extend the conventional LSR paradigm to implement higher-level logic architecture or typical digital electronic structures via carefully crafted coupling schemes. Further, we examine the intriguing possibility of obtaining reliable logic outputs from a noise-free bistable system, subject only to periodic forcing, and show that this system also yields a phenomenon analogous to LSR, termed Logical Vibrational Resonance (LVR), in an appropriate window of frequency and amplitude of the periodic forcing. Lastly, this approach is extended to realize morphable logic gates through the Logical Coherence Resonance (LCR) in excitable systems under the influence of noise. The results are verified with suitable circuit experiments, demonstrating the robustness of the LSR, LVR and LCR phenomena. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'.

Competitive interplay of repulsive coupling and cross-correlated noises in bistable systems.

The influence of noise on synchronization has potential impact on physical, chemical, biological, and engineered systems. Research on systems subject to common noise has demonstrated that noise can aid synchronization, as common noise imparts correlations on the sub-systems. In our work, we revisit this idea for a system of bistable dynamical systems, under repulsive coupling, driven by noises with varying degrees of cross correlation. This class of coupling has not been fully explored, and we show that it offers new counter-intuitive emergent behavior. Specifically, we demonstrate that the competitive interplay of noise and coupling gives rise to phenomena ranging from the usual synchronized state to the uncommon anti-synchronized state where the coupled bistable systems are pushed to different wells. Interestingly, this progression from anti-synchronization to synchronization goes through a domain where the system randomly hops between the synchronized and anti-synchronized states. The underlying basis for this striking behavior is that correlated noise preferentially enhances coherence, while the interactions provide an opposing drive to push the states apart. Our results also shed light on the robustness of synchronization obtained in the idealized scenario of perfectly correlated noise, as well as the influence of noise correlation on anti-synchronization. Last, the experimental implementation of our model using bistable electronic circuits, where we were able to sweep a large range of noise strengths and noise correlations in the laboratory realization of this noise-driven coupled system, firmly indicates the robustness and generality of our observations.

Ill-matched timescales in coupled systems can induce oscillation suppression.

We explore the behavior of two coupled oscillators, considering combinations of similar and dissimilar oscillators, with their intrinsic dynamics ranging from periodic to chaotic. We first investigate the coupling of two different real-world systems, namely, the chemical mercury beating heart oscillator and the electronic Chua oscillator, with the disparity in the timescales of the constituent oscillators. Here, we are considering a physical situation that is not commonly addressed: the coupling of sub-systems whose characteristic timescales are very different. Our findings indicate that the oscillations in coupled systems are quenched to oscillation death (OD) state, at sufficiently high coupling strength, when there is a large timescale mismatch. In contrast, phase synchronization occurs when their timescales are comparable. In order to further strengthen the concept, we demonstrate this timescale-induced oscillation suppression and phase synchrony through numerical simulations, with the disparity in the timescales serving as a tuning or control parameter. Importantly, oscillation suppression (OD) occurs for a significantly smaller timescale mismatch when the coupled oscillators are chaotic. This suggests that the inherent broad spectrum of timescales underlying chaos aids oscillation suppression, as the temporal complexity of chaotic dynamics lends a natural heterogeneity to the timescales. The diversity of the experimental systems and numerical models we have chosen as a test-bed for the proposed concept lends support to the broad generality of our findings. Last, these results indicate the potential prevention of system failure by small changes in the timescales of the constituent dynamics, suggesting a potent control strategy to stabilize coupled systems to steady states.

Emergence of extreme events in networks of parametrically coupled chaotic populations.

We consider a collection of populations modelled by the prototypical chaotic Ricker map, relevant to the population growth of species with non-overlapping generations. The growth parameter of each population patch is influenced by the local mean field of its neighbourhood, and we explore the emergent patterns in such a parametrically coupled network. In particular, we examine the dynamics and distribution of the local populations, as well as the total biomass. Our significant finding is the following: When the range of coupling is sufficiently large, namely, when enough neighbouring populations influence the growth rate of a population, the system yields remarkably large biomass values that are very far from the mean. These extreme events are relatively rare and uncorrelated in time. We also find that at any point in time, exceedingly large population densities emerge in a few patches, analogous to an extreme event in space. Thus, we suggest a new mechanism in coupled chaotic systems that naturally yield extreme events in both time and space.

Small-world networks exhibit pronounced intermittent synchronization.

We report the phenomenon of temporally intermittently synchronized and desynchronized dynamics in Watts-Strogatz networks of chaotic Rössler oscillators. We consider topologies for which the master stability function (MSF) predicts stable synchronized behaviour, as the rewiring probability (p) is tuned from 0 to 1. MSF essentially utilizes the largest non-zero Lyapunov exponent transversal to the synchronization manifold in making stability considerations, thereby ignoring the other Lyapunov exponents. However, for an N-node networked dynamical system, we observe that the difference in its Lyapunov spectra (corresponding to the N - 1 directions transversal to the synchronization manifold) is crucial and serves as an indicator of the presence of intermittently synchronized behaviour. In addition to the linear stability-based (MSF) analysis, we further provide global stability estimate in terms of the fraction of state-space volume shared by the intermittently synchronized state, as p is varied from 0 to 1. This fraction becomes appreciably large in the small-world regime, which is surprising, since this limit has been otherwise considered optimal for synchronized dynamics. Finally, we characterize the nature of the observed intermittency and its dominance in state-space as network rewiring probability (p) is varied.

Exploiting chaos for applications.

We discuss how understanding the nature of chaotic dynamics allows us to control these systems. A controlled chaotic system can then serve as a versatile pattern generator that can be used for a range of application. Specifically, we will discuss the application of controlled chaos to the design of novel computational paradigms. Thus, we present an illustrative research arc, starting with ideas of control, based on the general understanding of chaos, moving over to applications that influence the course of building better devices.

Noise tolerant spatiotemporal chaos computing.

We introduce and design a noise tolerant chaos computing system based on a coupled map lattice (CML) and the noise reduction capabilities inherent in coupled dynamical systems. The resulting spatiotemporal chaos computing system is more robust to noise than a single map chaos computing system. In this CML based approach to computing, under the coupled dynamics, the local noise from different nodes of the lattice diffuses across the lattice, and it attenuates each other's effects, resulting in a system with less noise content and a more robust chaos computing architecture.

Neuronal diversity can improve machine learning for physics and beyond.

Diversity conveys advantages in nature, yet homogeneous neurons typically comprise the layers of artificial neural networks. Here we construct neural networks from neurons that learn their own activation functions, quickly diversify, and subsequently outperform their homogeneous counterparts on image classification and nonlinear regression tasks. Sub-networks instantiate the neurons, which meta-learn especially efficient sets of nonlinear responses. Examples include conventional neural networks classifying digits and forecasting a van der Pol oscillator and physics-informed Hamiltonian neural networks learning Hénon-Heiles stellar orbits and the swing of a video recorded pendulum clock. Such learned diversity provides examples of dynamical systems selecting diversity over uniformity and elucidates the role of diversity in natural and artificial systems.

Influence of the Allee effect on extreme events in coupled three-species systems.

We considered the dynamics of two coupled three-species population patches by incorporating the Allee effect and focused on the onset of extreme events in the coupled system. First, we showed that the interplay between coupling and the Allee effect may change the nature of the dynamics, with regular periodic dynamics becoming chaotic in a range of Allee parameters and coupling strengths. Further, the growth in the vegetation population displays an explosive blow-up beyond a critical value of the coupling strength and Allee parameter. Most interestingly, we observed that beyond a threshold of the Allee parameter and coupling strength, the population densities of all three species exhibit a non-zero probability of yielding extreme events. The emergence of extreme events in the predator populations in the patches is the most prevalent, and the probability of obtaining large deviations in the predator populations is not affected significantly by either the coupling strength or the Allee effect. In the absence of the Allee effect, the prey population in the coupled system exhibits no extreme events for low coupling strengths, but yields a sharp increase in extreme events after a critical value of the coupling strength. The vegetation population in the patches displays a small finite probability of extreme events for strong enough coupling, only in the presence of the Allee effect. Last, we considered the influence of additive noise on the continued prevalence of extreme events. Very significantly, we found that noise suppresses the unbounded vegetation growth that was induced by a combination of the Allee effect and coupling. Further, we demonstrated that noise mitigates extreme events in all three populations, and beyond a noise level, we do not observe any extreme events in the system. This finding has important bearings on the potential observability of extreme events in natural and laboratory systems.

Machine-learning potential of a single pendulum.

Reservoir computing offers a great computational framework where a physical system can directly be used as computational substrate. Typically a "reservoir" is comprised of a large number of dynamical systems, and is consequently high dimensional. In this work, we use just a single simple low-dimensional dynamical system, namely, a driven pendulum, as a potential reservoir to implement reservoir computing. Remarkably we demonstrate, through numerical simulations as well as a proof-of-principle experimental realization, that one can successfully perform learning tasks using this single system. The underlying idea is to utilize the rich intrinsic dynamical patterns of the driven pendulum, especially the transient dynamics which has so far been an untapped resource. This allows even a single system to serve as a suitable candidate for a reservoir. Specifically, we analyze the performance of the single pendulum reservoir for two classes of tasks: temporal and nontemporal data processing. The accuracy and robustness of the performance exhibited by this minimal one-node reservoir in implementing these tasks strongly suggest an alternative direction in designing the reservoir layer from the point of view of efficient applications. Further, the simplicity of our learning system offers an opportunity to better understand the framework of reservoir computing in general and indicates the remarkable machine-learning potential of even a single simple nonlinear system.

Quenching of oscillations in a liquid metal via attenuated coupling.

In this work, we report a quenching of oscillations observed upon coupling two chemomechanical oscillators. Each one of these oscillators consists of a drop of liquid metal submerged in an oxidizing solution. These pseudoidentical oscillators have been shown to exhibit both periodic and aperiodic oscillatory behavior. In the experiments performed on these oscillators, we find that coupling two such oscillators via an attenuated resistive coupling leads the coupled system towards an oscillation quenched state. To further comprehend these experimental observations, we numerically explore and verify the presence of similar oscillation quenching in a model of coupled Hindmarsh-Rose (HR) systems. A linear stability analysis of this HR system reveals that attenuated coupling induces a change in eigenvalues of the relevant Jacobian, leading to stable quenched oscillation states. Additionally, the analysis yields a threshold of attenuation for oscillation quenching that is consistent with the value observed in numerics. So this phenomenon, demonstrated through experiments, as well as simulations and analysis of a model system, suggests a powerful natural mechanism that can potentially suppress periodic and aperiodic oscillations in coupled nonlinear systems.

A noise-assisted reprogrammable nanomechanical logic gate.

We present a nanomechanical device, operating as a reprogrammable logic gate, and performing fundamental logic functions such as AND/OR and NAND/NOR. The logic function can be programmed (e.g., from AND to OR) dynamically, by adjusting the resonator's operating parameters. The device can access one of two stable steady states, according to a specific logic function; this operation is mediated by the noise floor which can be directly adjusted, or dynamically "tuned" via an adjustment of the underlying nonlinearity of the resonator, i.e., it is not necessary to have direct control over the noise floor. The demonstration of this reprogrammable nanomechanical logic gate affords a path to the practical realization of a new generation of mechanical computers.

Enhancement of extreme events through the Allee effect and its mitigation through noise in a three species system.

We consider the dynamics of a three-species system incorporating the Allee Effect, focussing on its influence on the emergence of extreme events in the system. First we find that under Allee effect the regular periodic dynamics changes to chaotic. Further, we find that the system exhibits unbounded growth in the vegetation population after a critical value of the Allee parameter. The most significant finding is the observation of a critical Allee parameter beyond which the probability of obtaining extreme events becomes non-zero for all three population densities. Though the emergence of extreme events in the predator population is not affected much by the Allee effect, the prey population shows a sharp increase in the probability of obtaining extreme events after a threshold value of the Allee parameter, and the vegetation population also yields extreme events for sufficiently strong Allee effect. Lastly we consider the influence of additive noise on extreme events. First, we find that noise tames the unbounded vegetation growth induced by Allee effect. More interestingly, we demonstrate that stochasticity drastically diminishes the probability of extreme events in all three populations. In fact for sufficiently high noise, we do not observe any more extreme events in the system. This suggests that noise can mitigate extreme events, and has potentially important bearing on the observability of extreme events in naturally occurring systems.

Emergent noise-aided logic through synchronization.

In this article, we present a dynamical scheme to obtain a reconfigurable noise-aided logic gate that yields all six fundamental two-input logic operations, including the xor operation. The setup consists of two coupled bistable subsystems that are each driven by one subthreshold logic input signal, in the presence of a noise floor. The synchronization state of their outputs robustly maps to two-input logic operations of the driving signals, in an optimal window of noise and coupling strengths. Thus the interplay of noise, nonlinearity, and coupling leads to the emergence of logic operations embedded within the collective state of the coupled system. This idea is manifested using both numerical simulations and proof-of-principle circuit experiments. The regions in parameter space that yield reliable logic operations were characterized through a stringent measure of reliability, using both numerical and experimental data.

A coupled map lattice model for rheological chaos in sheared nematic liquid crystals.

A variety of complex fluids under shear exhibit complex spatiotemporal behavior, including what is now termed rheological chaos, at moderate values of the shear rate. Such chaos associated with rheological response occurs in regimes where the Reynolds number is very small. It must thus arise as a consequence of the coupling of the flow to internal structural variables describing the local state of the fluid. We propose a coupled map lattice model for such complex spatiotemporal behavior in a passively sheared nematic liquid crystal using local maps constructed so as to accurately describe the spatially homogeneous case. Such local maps are coupled diffusively to nearest and next-nearest neighbors to mimic the effects of spatial gradients in the underlying equations of motion. We investigate the dynamical steady states obtained as parameters in the map and the strength of the spatial coupling are varied, studying local temporal properties at a single site as well as spatiotemporal features of the extended system. Our methods reproduce the full range of spatiotemporal behavior seen in earlier one-dimensional studies based on partial differential equations. We report results for both the one- and two-dimensional cases, showing that spatial coupling favors uniform or periodically time-varying states, as intuitively expected. We demonstrate and characterize regimes of spatiotemporal intermittency out of which chaos develops. Our work indicates that similar simplified lattice models of the dynamics of complex fluids under shear should provide useful ways to access and quantify spatiotemporal complexity in such problems, in addition to representing a fast and numerically tractable alternative to continuum representations.

Introduction to focus issue: intrinsic and designed computation: information processing in dynamical systems--beyond the digital hegemony.

How dynamical systems store and process information is a fundamental question that touches a remarkably wide set of contemporary issues: from the breakdown of Moore's scaling laws--that predicted the inexorable improvement in digital circuitry--to basic philosophical problems of pattern in the natural world. It is a question that also returns one to the earliest days of the foundations of dynamical systems theory, probability theory, mathematical logic, communication theory, and theoretical computer science. We introduce the broad and rather eclectic set of articles in this Focus Issue that highlights a range of current challenges in computing and dynamical systems.

Chaogates: morphing logic gates that exploit dynamical patterns.

Chaotic systems can yield a wide variety of patterns. Here we use this feature to generate all possible fundamental logic gate functions. This forms the basis of the design of a dynamical computing device, a chaogate, that can be rapidly morphed to become any desired logic gate. Here we review the basic concepts underlying this and present an extension of the formalism to include asymmetric logic functions.