Spatiotemporal Chaos Research Articles

Extreme events following bifurcation to spatiotemporal chaos in a spatially extended microcavity laser

The occurrence of extreme events in a spatially extended microcavity laser has been recently reported [Selmi et al., Phys. Rev. Lett. 116, 013901 (2016)] to be correlated to emergence of spatiotemporal chaos. In this dissipative system, the role of spatial coupling through diffraction is essential to observe the onset of spatiotemporal complexity. We investigate further the formation mechanism of extreme events by comparing the statistical and dynamical analyses. Experimental measurements together with numerical simulations allow us to assign the quasiperiodicity mechanism as the route to spatiotemporal chaos in this system. Moreover, by investigating the fine structure of the maximum Lyapunov exponent, of the Lyapunov spectrum, and of the Kaplan-Yorke dimension of the chaotic attractor, we are able to deduce that intermittency plays a key role in the proportion of extreme events measured. We assign the observed mechanism of generation of extreme events to quasiperiodic extended spatiotemporal intermittency.

Terminating spiral wave and spatiotemporal chaos in cardiac tissues by using late sodium current

Most Na+ channels open transiently upon depolarization of cardiac cell membrane and then are quickly inactivated. However, some Na+ channels remain active, which generate the late sodium current during the action potential plateau. So far, late sodium current has been regarded as a relevant contributor to arrhythmias and its inhibition can suppress re-entrant and multifocal ventricular fibrillation so that its inhibition may become a novel therapeutic strategy to treat cardiac arrhythmias in the future. Therefore, how to inhibit late sodium current has received special attention. Since both the late sodium current and defibrillation shocks can lead to the increase of action potential duration, the late sodium current can be used to terminate ventricular fibrillation. However, the suppression of spiral wave and spatiotemporal chaos in cardiac tissues via late sodium current has been neglected. In this paper, we use the model of human heart to study the suppression of spiral wave and spatiotemporal chaos in two-dimensional cardiac tissue by generating late sodium current. We suggest that such a control strategy to induce late sodium current. The slow inactivation gate of sodium channel is clamped to 0.7 while the threshold voltage of corresponding fast inactivation gate is real-timely modulated. We first reduce the threshold voltage from 71.55 mV to 50.55 mV within the time interval T1, and then increase it from 50.55 mV to 71.55 mV within the time interval T2. When the threshold voltage returns to 71.55 mV, the changes of the relevant inactivation gates of sodium channel go back to normal dynamic state. Numerical simulation results show that when the control parameters are properly chosen, the control-induced late sodium current can effectively suppress spiral wave and spatiotemporal chaos even if there are some cardiac cells with spontaneous late sodium current. The advantage of the control scheme is that the control-induced late sodium current is small. The control duration is short because the spiral wave and spatiotemporal chaos disappear mainly due to the conduction obstacle. In a few cases, the spatiotemporal chaos disappears through the transition from spiral wave to target wave. We hope that these results may provide a new strategy to treat heart disease.

Emergent impacts of quadratic mortality on pattern formation in a predator–prey system

In this paper, we have investigated the effects of spatial diffusion in the pattern formation of a predator–prey system with quadratic mortality rate of the predator in the presence of habitat complexity. Using linear stability analysis, the regions of parameter space are plotted, and several kinds of instability regions are identified. Special attention has been given to investigate the selection of spatio-temporal patterns in the neighborhood of a critical parameter using the amplitude equation formalism. Choosing control parameter from the Turing space, the existence conditions for stable patterns are derived using the amplitude equations. Results obtained from theoretical analysis of amplitude equations agree very well with the numerical simulation results near the critical parameter value. Numerical simulations are done to show the existence of different spatio-temporal patterns. Spiral patterns are obtained in the suitable parameter region. It is shown that varying only the death rate of the predator, the spatio-temporal chaos can be controlled. Our investigations show that patterns are sensitive to the variation of habitat complexity and death rate of predator and patterns can be controlled by tuning the death rate of the predator.

Towards spin turbulence of light: Spontaneous disorder and chaos in cavity-polariton systems

Recent advances in nanophotonics have brought about coherent light sources with chaotic circular polarization; a low-dimensional chaotic evolution of optical spin was evidenced in laser diodes. Here we propose a mechanism that gives rise to light with a spatiotemporal spin chaos resembling turbulent states in hydrodynamics. The spin-chaotic radiation is emitted by exciton polaritons under resonant optical pumping in arbitrarily sized planar microcavities, including, as a limiting case, pointlike systems with only three degrees of freedom. The underlying mechanism originates in the interplay between spin symmetry breakdown and scattering into pairs of Bogolyubov excitations. As a practical matter, it opens up the way for spin modulation of light on the scale of picoseconds and micrometers.

Reply to “Comment on ‘Flow-induced arrest of spatiotemporal chaos and transition to a stationary pattern in the Gray-Scott model’ ”

We reply to the Comment of Berenstein etal. [preceding paper, Phys. Rev. E 94, 046201 (2016)10.1103/PhysRevE.94.046201] on our article [D. Das, Phys.Rev. E 92, 052914 (2015)10.1103/PhysRevE.92.052914] about the effect of streamlined flow on the dynamics of the Gray-Scott model characterized by wave-induced spatiotemporal chaos.

Comment on "Flow-induced arrest of spatiotemporal chaos and transition to a stationary pattern in the Gray-Scott model".

In this Comment, we review the results of pattern formation in a reaction-diffusion-advection system following the kinetics of the Gray-Scott model. A recent paper by Das [Phys. Rev. E 92, 052914 (2015)10.1103/PhysRevE.92.052914] shows that spatiotemporal chaos of the intermittency type can disappear as the advective flow is increased. This study, however, refers to a single point in the space of kinetic parameters of the original Gray-Scott model. Here we show that the wealth of patterns increases substantially as some of these parameters are changed. In addition to spatiotemporal intermittency, defect-mediated turbulence can also be found. In all cases, however, the chaotic behavior is seen to disappear as the advective flow is increased, following a scenario similar to what was reported in our earlier work [I. Berenstein and C. Beta, Phys. Rev. E 86, 056205 (2012)10.1103/PhysRevE.86.056205] as well as by Das. We also point out that a similar phenomenon can be found in other reaction-diffusion-advection models, such as the Oregonator model for the Belousov-Zhabotinsky reaction under flow conditions.

Relaxation with long-period oscillation in defect turbulence of planar nematic liquid crystals.

Through experiments, we studied defect turbulence, a type of spatiotemporal chaos in planar systems of nematic liquid crystals, to clarify the chaotic advection of weak turbulence. In planar systems of large aspect ratio, structural relaxation, which is characterized by the dynamic structure factor, exhibits a long-period oscillation that is described well by a combination of a simple exponential relaxation and underdamped oscillation. The simple relaxation arises as a result of the roll modulation while the damped oscillation is manifest in the repetitive gliding of defect pairs in a local area. Each relaxation is derived analytically by the projection operator method that separates turbulent transport into a macroscopic contribution and fluctuations. The analysis proposes that the two relaxations are not correlated. The nonthermal fluctuations of defect turbulence are consequently separated into two independent Markov processes. Our approach sheds light on diversity and universality from a unified viewpoint for weak turbulence.

Observations of spatiotemporal instabilities of temporal cavity solitons

Localized dissipative structures (LDS) have been predicted to display a rich array of instabilities, yet systematic experimental studies have remained scarce. We have used a synchronously-driven optical fiber ring resonator to experimentally study LDS instabilities in the strong-driving regime of the AC-driven nonlinear Schr\"odinger equation (also known as the Lugiato-Lefever model). Through continuous variation of a single control parameter, we have observed a string of theoretically predicted instability modes, including irregular oscillations and chaotic collapses. Beyond a critical point, we observe behaviour reminiscent of a phase transition: LDSs trigger localized domains of spatiotemporal chaos that invade the surrounding homogeneous state. Our findings directly confirm a number of theoretical predictions, and they highlight that complex LDS instabilities can play a role in experimental systems.

Correlation analysis of the coherence-incoherence transition in a ring of nonlocally coupled logistic maps.

We present numerical results for a set of bifurcations occurring at the transition from complete chaotic synchronization to spatio-temporal chaos in a ring of nonlocally coupled chaotic logistic maps. The regularities are established for the evolution of cross-correlations of oscillations in the network elements at the bifurcations related to the coupling strength variation. We reveal the distinctive features of cross-correlations for phase and amplitude chimera states. It is also shown that the effect of time intermittency between the amplitude and phase chimeras can be realized in the considered ensemble.

Design of synchronization technique for uncertain discrete network group with diverse structures

Abstract In this work, we design a novel synchronization technique to realize the synchronization of the network group which is constituted by some uncertain discrete networks with diverse structures. Based on Lyapunov theorem, the selection principle of the control inputs and identification law of uncertain parameters in networks are determined, and the synchronization conditions of network group are obtained. Finally, numerical simulations using one-dimensional convective equations with spatiotemporal chaos behaviors illustrate the performance of the synchronization scheme. The research results show that our synchronization technique can be suitable for the network connecting arbitrarily, and not only the network number but also node number in each network can also be chosen freely.

Applying a modified version of Lyapunov exponent for cancer diagnosis in biomedical images: the case of breast mammograms

Recently, there has been a great interest in the application of Lyapunov exponents for calculation of chaos levels in dynamical systems. Accordingly, this study aims at presenting two new methods for utilizing Lyapunov exponents to evaluate the spatiotemporal chaos in various images. Further, early detection of cancerous tumors could be obtained by measuring the chaotic indices in biomedical images. Unlike the available systems described by partial differential equations, the proposed method employs a number of interactive dynamic variables for image modeling. Since the Lyapunov exponents cannot be applied to such systems, the image model should be modified. The mean Lyapunov exponent is defined as a chaotic index for measuring the contour borders irregularities in images to detect benign or malignant tumors. Moreover, a two-dimensional mean Lyapunov exponent is incorporated to identify irregularities existing in each axis of the targeted images. Experiments on a set of region of interest in breast mammogram images yielded a sensitivity of 95 % and a specificity of 97.3 % and verified the remarkable precision of the proposed methods in classifying of breast lesions obtained from breast mammogram images.

Controlling and synchronizing the spatiotemporal chaos of photorefractive ring oscillators with coupling

We present the control and synchronization of spatiotemporal chaos in the photo-refractive ring oscillator systems with coupling technology. First, we realize the synchronization of spatiotemporal chaos in the two photorefractive ring oscillator systems via mutual coupling by choosing a suitable coupling strength. With the mutual coupling strength enlarging, the two mutual coupling photorefractive ring oscillator systems are controlled into periodic state, period number differs on account of the coupling strength and lattice coordinates. By increasing the coupling strength, the photorefractive ring oscillator is converted into period 8, subsequently it is converted into periods 4 and 2, periodic synchronization of the photorefractive ring oscillator systems is achieved at the same time. Calculation results show that period 1 is impossible by mutual coupling technology. Then, we investigate the influence of noise and parameter deviation on chaotic synchronization. We find that mutual coupling chaotic synchronization method can synchronize two chaotic systems with the weak noise and parameter deviation and has very good robustness. Given that the weak noise and parameter deviation have a slight effect on synchronization. Furthermore, we investigate two dimension control and synchronization of spatiotemporal chaos in the photorefractive ring osillator systems with coupling technology and get successful results. Mutual coupling technology is suitable in practical photorefractive ring oscillator systems.

Complex patterns in a space- and time-discrete predator-prey model with Beddington-DeAngelis functional response

The spatial pattern formation of predator-prey systems is an important issue widely concerned. In this research, we address this issue by developing a new space- and time-discrete predator-prey model, with predation relationship described by Beddington-DeAngelis functional response. The discrete model is given by a coupled map lattice, taking a nonlinear relationship between predator-prey “reaction” stage and dispersal stage. Through analysis of Turing instability and Hopf instability for the discrete model, the parametric conditions for pattern formation are determined. Numerical simulations reveal a surprising variety of spatiotemporal patterns, including regular and irregular patterns of spots, stripes, labyrinth, gaps, mosaics, spirals, circles, and many intermediate patterns in-between. These patterns cover a majority of predator-prey pattern types recorded in literature. Besides, the discrete model predicts the occurrence of spatiotemporal chaos, which is responsible for the formation of irregular patterns. This research demonstrates that the nonlinear mechanisms of the discrete model better capture the complexity of pattern formation of predator-prey systems.

Delayed-feedback chimera states: Forced multiclusters and stochastic resonance

A nonlinear oscillator model with negative time-delayed feedback is studied numerically under external deterministic and stochastic forcing. It is found that in the unforced system complex partial synchronization patterns like chimera states as well as salt-and-pepper–like solitary states arise on the route from regular dynamics to spatio-temporal chaos. The control of the dynamics by external periodic forcing is demonstrated by numerical simulations. It is shown that one-cluster and multi-cluster chimeras can be achieved by adjusting the external forcing frequency to appropriate resonance conditions. If a stochastic component is superimposed to the deterministic external forcing, chimera states can be induced in a way similar to stochastic resonance, they appear, therefore, in regimes where they do not exist without noise.

Optical textures: characterizing spatiotemporal chaos.

Macroscopic systems subjected to injection and dissipation of energy can exhibit complex spatiotemporal behaviors as result of dissipative self-organization. Here, we report a one- and two-dimensional pattern forming setup, which exhibits a transition from stationary patterns to spatiotemporal chaotic textures, based on a nematic liquid crystal layer with spatially modulated input beam and optical feedback. Using an adequate projection of spatiotemporal diagrams, we determine the largest Lyapunov exponent. Jointly, this exponent and Fourier transform allow us to distinguish between spatiotemporal chaos and amplitude turbulence concepts, which are usually merged.

Towards probabilistic synchronisation of local controllers

ABSTRACTThe traditional use of global and centralised control methods fails for large, complex, noisy and highly connected systems, which typify many real-world industrial and commercial systems. This paper provides an efficient bottom-up design of distributed control in which many simple components communicate and cooperate to achieve a joint system goal. Each component acts individually so as to maximise personal utility whilst obtaining probabilistic information on the global system merely through local message-passing. This leads to an implied scalable and collective control strategy for complex dynamical systems, without the problems of global centralised control. Robustness is addressed by employing a fully probabilistic design, which can cope with inherent uncertainties, can be implemented adaptively and opens a systematic rich way to information sharing. This paper opens the foreseen direction and inspects the proposed design on a linearised version of coupled map lattice with spatio-temporal chaos. A version close to linear quadratic design gives an initial insight into possible behaviours of such networks.

Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection.

We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.

Extreme events induced by spatiotemporal chaos in experimental optical patterns.

Extreme events such as rogue waves are often associated with the merging of coherent structures. We report on experimental results in the physics of extreme events emerging in a liquid-crystal light valve subjected to optical feedback, and we establish the relation of this phenomenon with the appearance of spatiotemporal chaos. This system, under particular conditions, exhibits stationary roll patterns that can be destabilized into quasi-periodic and chaotic textures when control parameters are properly modified. We have identified the parameter regions where extreme fluctuations of the amplitude can emerge and established their origin through its direct relation with the experimental largest Lyapunov exponents, the proportion of extreme events, and the normed kurtosis.

Quasiperiodicity route to chaos in cardiac conduction model

It has been suggested that cardiac arrhythmias are instances of chaos. In particular that the ventricular fibrillation is a form of spatio-temporal chaos that arises from normal rhythm through a quasi-periodicity or Ruelle-Takens-Newhouse route to chaos. In this work, we modify the heterogeneous oscillator model of cardiac conduction system proposed in Ref. [Ryzhii E, Ryzhii M. A heterogeneous coupled oscillator model for simulation of ECG signals. Comput Meth Prog Bio 2014;117(1):40–49. doi:10.1016/j.cmpb.2014.04.009.], by including an ectopic pacemaker that stimulates the ventricular muscle to model arrhythmias. With this modification, the transition from normal rhythm to ventricular fibrillation is controlled by a single parameter. We show that this transition follows the so-called torus of quasi-periodic route to chaos, as verified by using numerical tools such as power spectrum and largest Lyapunov exponent.

Two-particle dispersion in weakly turbulent thermal convection

We present results from a numerical study of particle dispersion in the weakly nonlinear regime of Rayleigh–Bénard convection of a fluid with Prandtl number around unity, where bi-stability between ideal straight convection rolls and weak turbulence in the form of spiral defect chaos exists. While Lagrangian pair statistics has become a common tool for studying fully developed turbulent flows at high Reynolds numbers, we show that key characteristics of mass transport can also be found in convection systems that show no or weak turbulence. Specifically, for short times, we find an interval of t3-scaling of pair dispersion, which we explain quantitatively with the interplay of advection and diffusion. For long times we observe diffusion-like dispersion of particles that becomes independent of the individual particles’ stochastic movements. The spreading rate is found to depend on the degree of spatio-temporal chaos.