Spatially-extended Systems Research Articles

Time crystal dynamics in a weakly modulated stochastic time delayed system

Time crystal oscillations in interacting, periodically driven many-particle systems are highly regular oscillations that persist for long periods of time, are robust to perturbations, and whose frequency differs from the frequency of the driving signal. Making use of underlying similarities of spatially-extended systems and time-delayed systems (TDSs), we present an experimental demonstration of time-crystal-like behavior in a stochastic, weakly modulated TDS. We consider a semiconductor laser near threshold with delayed feedback, whose output intensity shows abrupt spikes at irregular times. When the laser current is driven with a small-amplitude periodic signal we show that the interaction of delayed feedback and modulation can generate long-range regularity in the timing of the spikes, which lock to the modulation and, despite the presence of noise, remain in phase over thousands of modulation cycles. With pulsed modulation we find harmonic and subharmonic locking, while with sinusoidal modulation, we find only subharmonic locking, which is a characteristic feature of time-crystal behavior.

Universality of noise-induced resilience restoration in spatially-extended ecological systems

Many systems may switch to an undesired state due to internal failures or external perturbations, of which critical transitions toward degraded ecosystem states are prominent examples. Resilience restoration focuses on the ability of spatially-extended systems and the required time to recover to their desired states under stochastic environmental conditions. The difficulty is rooted in the lack of mathematical tools to analyze systems with high dimensionality, nonlinearity, and stochastic effects. Here we show that nucleation theory can be employed to advance resilience restoration in spatially-embedded ecological systems. We find that systems may exhibit single-cluster or multi-cluster phases depending on their sizes and noise strengths. We also discover a scaling law governing the restoration time for arbitrary system sizes and noise strengths in two-dimensional systems. This approach is not limited to ecosystems and has applications in various dynamical systems, from biology to infrastructural systems.

Lag-invariant detection of interactions in spatially-extended systems using linear inverse modeling.

Measurements on physical systems result from the systems' activity being converted into sensor measurements by a forward model. In a number of cases, inversion of the forward model is extremely sensitive to perturbations such as sensor noise or numerical errors in the forward model. Regularization is then required, which introduces bias in the reconstruction of the systems' activity. One domain in which this is particularly problematic is the reconstruction of interactions in spatially-extended complex systems such as the human brain. Brain interactions can be reconstructed from non-invasive measurements such as electroencephalography (EEG) or magnetoencephalography (MEG), whose forward models are linear and instantaneous, but have large null-spaces and high condition numbers. This leads to incomplete unmixing of the forward models and hence to spurious interactions. This motivated the development of interaction measures that are exclusively sensitive to lagged, i.e. delayed interactions. The drawback of such measures is that they only detect interactions that have sufficiently large lags and this introduces bias in reconstructed brain networks. We introduce three estimators for linear interactions in spatially-extended systems that are uniformly sensitive to all lags. We derive some basic properties of and relationships between the estimators and evaluate their performance using numerical simulations from a simple benchmark model.

Complex interactions can create persistent fluctuations in high-diversity ecosystems.

When can ecological interactions drive an entire ecosystem into a persistent non-equilibrium state, where many species populations fluctuate without going to extinction? We show that high-diversity spatially heterogeneous systems can exhibit chaotic dynamics which persist for extremely long times. We develop a theoretical framework, based on dynamical mean-field theory, to quantify the conditions under which these fluctuating states exist, and predict their properties. We uncover parallels with the persistence of externally-perturbed ecosystems, such as the role of perturbation strength, synchrony and correlation time. But uniquely to endogenous fluctuations, these properties arise from the species dynamics themselves, creating feedback loops between perturbation and response. A key result is that fluctuation amplitude and species diversity are tightly linked: in particular, fluctuations enable dramatically more species to coexist than at equilibrium in the very same system. Our findings highlight crucial differences between well-mixed and spatially-extended systems, with implications for experiments and their ability to reproduce natural dynamics. They shed light on the maintenance of biodiversity, and the strength and synchrony of fluctuations observed in natural systems.

Convolutional autoencoder and conditional random fields hybrid for predicting spatial-temporal chaos.

We present an approach for data-driven prediction of high-dimensional chaotic time series generated by spatially-extended systems. The algorithm employs a convolutional autoencoder for dimension reduction and feature extraction combined with a probabilistic prediction scheme operating in the feature space, which consists of a conditional random field. The future evolution of the spatially-extended system is predicted using a feedback loop and iterated predictions. The excellent performance of this method is illustrated and evaluated using Lorenz-96 systems and Kuramoto-Sivashinsky equations of different size generating time series of different dimensionality and complexity.

Turing patterns via pinning control in the simplest memristive cellular nonlinear networks.

Complex patterns are commonly retrieved in spatially-extended systems formed by coupled nonlinear dynamical units. In particular, Turing patterns have been extensively studied investigating mathematical models pertaining to different fields, such as chemistry, physics, biology, mechanics, and electronics. In this paper, we focus on the emergence of Turing patterns in memristive cellular nonlinear networks by means of spatial pinning control. The circuit architecture is made by coupled units formed by only two elements, namely, a capacitor and a memristor. The analytical conditions for which Turing patterns can be derived in the proposed architecture are discussed in order to suitably design the circuit parameters. In particular, we derive the conditions on the density of the controlled nodes for which a Turing pattern is globally generated. Finally, it is worth to note that the proposed architecture can be considered as the simplest ideal electronic circuit able to undergo Turing instability and give rise to pattern formation.

Fractional Kuramoto–Sivashinsky equation with power law and stretched Mittag-Leffler kernel

The presence of a complex spatio-temporal behavior in spatially extended systems (SES) as a result of several mechanisms that interact non-linearly with other nearby has attracted a lot of attention in recent decades. A well-known example of SES is the Kuramoto–Sivashinsky (KS) equation. In the search for a broader perspective of some unusual irregularities observed in the context of phase turbulence in the reaction–diffusion systems, the propagation of the wrinkled flame front and the unstable drift waves driven by the collision of electrons in a Tokamak, we explored the possibility of extending the analysis of the KS equation with three perturbation levels using the conceptions of fractional differentiation with non-local and non-singular kernel. We use the fractional order operator of Atangana–Baleanu in the sense of Liouville–Caputo (ABC) to set the fractional model of KS that we analyze in this article. We prove existence and uniqueness of a continuous solution to fractional KS model, using the fixed-point theorem and the Picard–Lindelöf approach, provided conditions on the perturbation parameters and the order of the fractional operator. In addion, using the theory of fixed point, we present the stability of the iterative method generated by the Picard–Lindelöf approach. Finally, we presented the approximate analytical solution of the fractional KS equation using the homotopy perturbation transform method. Some numerical simulations are carried out for illustrating the results obtained.

Excitable Wave Patterns in Temporal Systems with Two Long Delays and their Observation in a Semiconductor Laser Experiment.

Excitable waves arise in many spatially extended systems of either a biological, chemical, or physical nature due to the interplay between local reaction and diffusion processes. Here we demonstrate that similar phenomena are encoded in the time dynamics of an excitable system with two, hierarchically long delays. The transition from 1D localized structures to curved wave segments is experimentally observed in an excitable semiconductor laser with two feedback loops and reproduced by numerical simulations of a prototypical model. While closely related to those found in 2D excitable media, wave patterns in delayed systems exhibit unobserved features originating from causality related constraints. An appropriate dynamical representation of the data uncovers these phenomena and permits us to interpret them as the result of an effective 2D advection-reaction-diffusion process.

Velocity-dependent Lyapunov exponents in many-body quantum, semiclassical, and classical chaos

The exponential growth or decay with time of the out-of-time-order commutator (OTOC) is one widely used diagnostic of many-body chaos in spatially-extended systems. In studies of many-body classical chaos, it has been noted that one can define a velocity-dependent Lyapunov exponent, $\lambda({\bf v})$, which is the growth or decay rate along "rays" at that velocity. We examine the behavior of $\lambda({\bf v})$ for a variety of many-body systems, both chaotic and integrable. The so-called light cone for the spreading of operators is defined by $\lambda({\bf \hat n}v_B({\bf \hat n}))=0$, with a generally direction-dependent "butterfly speed" $v_B({\bf \hat n})$. In spatially local systems, $\lambda(v)$ is negative outside the light cone where it takes the form $\lambda(v) \sim -(v-v_B)^{\alpha}$ near $v_b$, with the exponent $\alpha$ taking on various values over the range of systems we examine. The regime inside the light cone with positive Lyapunov exponents may only exist for classical, semi-classical or large-$N$ systems, but not for "fully quantum" chaotic systems with strong short-range interactions and local Hilbert space dimensions of order one.

Delayed loss of stability due to the slow passage through Hopf bifurcations in reaction-diffusion equations.

This article presents the delayed loss of stability due to slow passage through Hopf bifurcations in reaction-diffusion equations with slowly-varying parameters, generalizing a well-known result about delayed Hopf bifurcations in analytic ordinary differential equations to spatially-extended systems. We focus on the Hodgkin-Huxley partial differential equation (PDE), the cubic Complex Ginzburg-Landau PDE as an equation in its own right, the Brusselator PDE, and a spatially-extended model of a pituitary clonal cell line. Solutions which are attracted to quasi-stationary states (QSS) sufficiently before the Hopf bifurcations remain near the QSS for long times after the states have become repelling, resulting in a significant delay in the loss of stability and the onset of oscillations. Moreover, the oscillations have large amplitude at onset, and may be spatially homogeneous or inhomogeneous. Space-time boundaries are identified that act as buffer curves beyond which solutions cannot remain near the repelling QSS, and hence before which the delayed onset of oscillations must occur, irrespective of initial conditions. In addition, a method is developed to derive the asymptotic formulas for the buffer curves, and the asymptotics agree well with the numerically observed onset in the Complex Ginzburg-Landau (CGL) equation. We also find that the first-onset sites act as a novel pulse generation mechanism for spatio-temporal oscillations.

Spatiotemporal canards in neural field equations.

Canards are special solutions to ordinary differential equations that follow invariant repelling slow manifolds for long time intervals. In realistic biophysical single-cell models, canards are responsible for several complex neural rhythms observed experimentally, but their existence and role in spatially extended systems is largely unexplored. We identify and describe a type of coherent structure in which a spatial pattern displays temporal canard behavior. Using interfacial dynamics and geometric singular perturbation theory, we classify spatiotemporal canards and give conditions for the existence of folded-saddle and folded-node canards. We find that spatiotemporal canards are robust to changes in the synaptic connectivity and firing rate. The theory correctly predicts the existence of spatiotemporal canards with octahedral symmetry in a neural field model posed on the unit sphere.

Universal scaling of Lyapunov-exponent fluctuations in space-time chaos

Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase space. A recent numerical study of spatially extended systems has revealed that the diffusion coefficient D of the Lyapunov exponents (LEs) exhibits a nontrivial scaling behavior, D(L)~L(-γ), with the system size L. Here, we show that the wandering exponent γ can be expressed in terms of the roughening exponents associated with the corresponding "Lyapunov surface." Our theoretical predictions are supported by the numerical analysis of several spatially extended systems. In particular, we find that the wandering exponent of the first LE is universal: in view of the known relationship with the Kardar-Parisi-Zhang equation, γ can be expressed in terms of known critical exponents. Furthermore, our simulations reveal that the bulk of the spectrum exhibits a clearly different behavior and suggest that it belongs to a possibly unique universality class, which has, however, yet to be identified.

QUANTIFYING EMERGENCE IN TERMS OF PERSISTENT MUTUAL INFORMATION

We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the past history of a system and its evolution significantly later in the future. This quantifies how much past observations enable long-term prediction, which we propose as the primary signature of (Strong) Emergent Behavior. The key feature of our definition of PMI is the omission of an interval of "present" time, so that the mutual information between close times is excluded: this renders PMI robust to superposed noise or chaotic behavior or graininess of data, distinguishing it from a range of established Complexity Measures. For the logistic map, we compare predicted with measured long-time PMI data. We show that measured PMI data captures not just the period doubling cascade but also the associated cascade of banded chaos, without confusion by the overlayer of chaotic decoration. We find that the standard map has apparently infinite PMI, but with well-defined fractal scaling which we can interpret in terms of the relative information codimension. Whilst our main focus is in terms of PMI over time, we can also apply the idea to PMI across space in spatially-extended systems as a generalization of the notion of ordered phases.

Spatially extended populations reproducing logistic map

Abstract We discuss here the conditions that the spatially extended systems (SES) must satisfy to reproduce the logistic map. To address this dilemma we define a 2-D coupled map lattice with a local rule mimicking the logistic formula. We show that for growth rates of k⩽k ∞ (k ∞ is the accumulation point) the global evolution of the system exactly reproduces the cascade of period doubling bifurcations. However, for k > k ∞, instead of chaotic modes, the cascade of period halving bifurcations is observed. Consequently, the microscopic states at the lattice nodes resynchronize producing dynamically changing spatial patterns. By downscaling the system and assuming intense mobility of individuals over the lattice, the spatial correlations can be destroyed and the local rule remains the only factor deciding the evolution of the whole colony. We found the class of “atomistic” rules for which uncorrelated spatially extended population matches the logistic map both for pre-chaotic and chaotic modes. We concluded that the global logistic behavior can be expected for a spatially extended colony with high mobility of individuals whose microscopic behavior is governed by a specific semi-logistic rule in the closest neighborhood. Conversely, the populations forming dynamically changing spatial clusters behave in a different way than the logistic model and reproduce at least the steady-state fragment of the logistic map.

Toward homochiral protocells in noncatalytic peptide systems.

The activation-polymerization-epimerization-depolymerization (APED) model of Plasson et al. has recently been proposed as a mechanism for the evolution of homochirality on prebiotic Earth. The dynamics of the APED model in two-dimensional spatially-extended systems is investigated for various realistic reaction parameters. It is found that the APED system allows for the formation of isolated homochiral proto-domains surrounded by a racemate. A diffusive slowdown of the APED network induced, for example, through tidal motion or evaporating pools and lagoons leads to the stabilization of homochiral bounded structures as expected in the first self-assembled protocells.

Automatic filters for the detection of coherent structure in spatiotemporal systems

Most current methods for identifying coherent structures in spatially extended systems rely on prior information about the form which those structures take. Here we present two approaches to automatically filter the changing configurations of spatial dynamical systems and extract coherent structures. One, local sensitivity filtering, is a modification of the local Lyapunov exponent approach suitable to cellular automata and other discrete spatial systems. The other, local statistical complexity filtering, calculates the amount of information needed for optimal prediction of the system's behavior in the vicinity of a given point. By examining the changing spatiotemporal distributions of these quantities, we can find the coherent structures in a variety of pattern-forming cellular automata, without needing to guess or postulate the form of that structure. We apply both filters to elementary and cyclical cellular automata (ECA and CCA) and find that they readily identify particles, domains, and other more complicated structures. We compare the results from ECA with earlier ones based upon the theory of formal languages and the results from CCA with a more traditional approach based on an order parameter and free energy. While sensitivity and statistical complexity are equally adept at uncovering structure, they are based on different system properties (dynamical and probabilistic, respectively) and provide complementary information.

Subbasins, portals, and mazes: Transients in high dimensions

Transients in spatially-extended systems can be so long that the asymptotic invariant measure, or attractor, is irrelevant to the observed behavior. Even if the dynamics admits no local information production, a transient can appear quite complex spatially or temporally unpredictable. In such situations, the observed complexity depends on the basin structure in a high-dimensional state space. Statistical methods are used to elucidate the geometry of two broad classes of transient behavior: stationary and nonstationary.