Mechanisms of extensive spatiotemporal chaos in Rayleigh-Benard convection

Annihilation dynamics during spiral defect chaos revealed by particle models

Pair-annihilation events are ubiquitous in a variety of spatially extended systems and are often studied using computationally expensive simulations. Here, we develop an approach in which we simulate the pair-annihilation of spiral wave tips in cardiac models using a computationally efficient particle model. Spiral wave tips are represented as particles with dynamics governed by diffusive behavior and short-ranged attraction. The parameters for diffusion and attraction are obtained by comparing particle motion to the trajectories of spiral wave tips in cardiac models during spiral defect chaos. The particle model reproduces the annihilation rates of the cardiac models and can determine the statistics of spiral wave dynamics, including its mean termination time. We show that increasing the attraction coefficient sharply decreases the mean termination time, making it a possible target for pharmaceutical intervention.

The center for nonlinear studies: A personal history

The Center for Nonlinear Studies (CNLS) was an integral part of my scientific career starting as a Postdoctoral Fellow in 1983 up to my tenure as CNLS Director from 2004 to 2015. As such, I experienced a number of scientific phases of CNLS through almost four decades of foundation, evolution, and transition. Throughout this entire interval, the inspiration and influence of David Campbell guided my way. A proper history of CNLS encompassing all of the many contributors to the CNLS story is beyond my means or purpose here. Instead, I present the history as I experienced it. I emphasize the main scientific accomplishments achieved at CNLS over more than 40 years, but I will also attempt to describe and quantify the attributes that made and continue to make the Center for Nonlinear Studies a special institution of remarkable impact and longevity. Throughout its existence, CNLS owes much to the enduring legacy of David Campbell who laid down the foundations and operating principles that have made it so successful.

Characterization of stability of dynamic particle ensemble systems using topological data analysis

Holes are ubiquitous structures in phase space, and their time evolution could indicate an instability in the dynamics of the system. However, the properties of these holes are difficult to study directly due to their theoretical complexity and lack of computational tools. This study proposes the use of persistent homology (PH), a technique from topological data analysis, as a computational tool for analyzing the properties of these phase-space holes, or more formally the H1 homology class according to PH. Initially, by using a toy data set, it is shown that the time evolution and the growth rate of a H1 class in phase space could be obtained by PH. For further validation, PH is applied to particle ensemble systems, such as the Hamiltonian flow and the two-stream instability (TSI). Both the stable case, where no H1 forms, and the unstable case, where H1 forms, were analyzed. It was shown that PH can distinguish between the stable and unstable cases purely from the phase-space time evolution plots. In unstable TSI, the PH also distinguished the transition of the H1 class from linear to non-linear growth. The growth rate, thus, obtained is in excellent agreement with the growth rate of the particle energy in the TSI system.

Using covariant Lyapunov vectors to quantify high-dimensional chaos with a conservation law

We explore the high-dimensional chaos of a one-dimensional lattice of diffusively coupled tent maps using the covariant Lyapunov vectors (CLVs). We investigate the connection between the dynamics of the maps in the physical space and the dynamics of the covariant Lyapunov vectors and covariant Lyapunov exponents that describe the direction and growth (or decay) of small perturbations in the tangent space. We explore the tangent space splitting into physical and transient modes and find that the splitting persists for all of the conditions we explore. In general, the leading CLVs are highly localized in space and the CLVs become less localized with increasing Lyapunov index. We consider the dynamics with a conservation law whose strength is controlled by a parameter that can be continuously varied. Our results indicate that a conservation law delocalizes the spatial variation of the CLVs. We find that when a conservation law is present, the leading CLVs are entangled with fewer of their neighboring CLVs than in the absence of a conservation law.

Markov modeling of phase singularity interaction effects in human atrial and ventricular fibrillation

Atrial and ventricular fibrillation (AF/VF) are characterized by the repetitive regeneration of topological defects known as phase singularities (PSs). The effect of PS interactions has not been previously studied in human AF and VF. We hypothesized that PS population size would influence the rate of PS formation and destruction in human AF and VF, due to increased inter-defect interaction. PS population statistics were studied in computational simulations (Aliev-Panfilov), human AF and human VF. The influence of inter-PS interactions was evaluated by comparison between directly modeled discrete-time Markov chain (DTMC) transition matrices of the PS population changes, and M/M/∞ birth-death transition matrices of PS dynamics, which assumes that PS formations and destructions are effectively statistically independent events. Across all systems examined, PS population changes differed from those expected with M/M/∞. In human AF and VF, the formation rates decreased slightly with PS population when modeled with the DTMC, compared with the static formation rate expected through M/M/∞, suggesting new formations were being inhibited. In human AF and VF, the destruction rates increased with PS population for both models, with the DTMC rate increase exceeding the M/M/∞ estimates, indicating that PS were being destroyed faster as the PS population grew. In human AF and VF, the change in PS formation and destruction rates as the population increased differed between the two models. This indicates that the presence of additional PS influenced the likelihood of new PS formation and destruction, consistent with the notion of self-inhibitory inter-PS interactions.

The physics of heart rhythm disorders

The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.

Role of interatrial conduction in atrial fibrillation. Mechanistic insights from renewal theory-based fibrillatory dynamic analysis

Background

Interatrial conduction has been postulated to play an important role in atrial fibrillation (AF). The pathways involved in interatrial conduction during AF remain incompletely defined.

Objective

We recently showed physiological assessment of fibrillatory dynamics could be performed using renewal theory, which determines rates of phase singularity formation (λ_f) and destruction (λ_d). Using the renewal approach, we aimed to understand the role of the interatrial septum and other electrically coupled regions during AF.

Method

RENEWAL-AF is a prospective multicenter observational study recruiting AF ablation patients (ACTRN 12619001172190). We studied unipolar electrograms obtained from 16 biatrial locations prior to ablation using a 16-electrode Advisor HD Grid catheter. Renewal rate constants λ_f and λ_d were calculated, and the relationships between these rate constants in regions of interatrial connectivity were examined.

Results

Forty-one AF patients (28.5% female) were recruited. A positive linear correlation was observed between λ_f and λ_d (1) across the interatrial septum (λ_f r² = 0.5, P < .001, λ_d r² = 0.45, P < .001), (2) in regions connected by the Bachmann bundle (right atrial appendage–left atrial appendage λ_f r² = 0.29, P = .001; λ_d r² = 0.2, P = .008), and (3) across the inferior interatrial routes (cavotricuspid isthmus–left atrial septum λ_f r² = 0.67, P < .001; λ_d r² = 0.55, P < .001). Persistent AF status and left atrial volume were found to be important effect modifiers of the degree of interatrial renewal rate statistical correlation.

Conclusion

Our findings support the role of interseptal statistically determined electrical disrelation in sustaining AF. Additionally, renewal theory identified preferential conduction through specific interatrial pathways during fibrillation. These findings may be of importance in identifying clinically significant targets for ablation in AF patients.

The inspection paradox: An important consideration in the evaluation of rotor lifetimes in cardiac fibrillation

Background and Objective: Renewal theory is a statistical approach to model the formation and destruction of phase singularities (PS), which occur at the pivots of spiral waves. A common issue arising during observation of renewal processes is an inspection paradox, due to oversampling of longer events. The objective of this study was to characterise the effect of a potential inspection paradox on the perception of PS lifetimes in cardiac fibrillation. Methods: A multisystem, multi-modality study was performed, examining computational simulations (Aliev-Panfilov (APV) model, Courtmanche-Nattel model), experimentally acquired optical mapping Atrial and Ventricular Fibrillation (AF/VF) data, and clinically acquired human AF and VF. Distributions of all PS lifetimes across full epochs of AF, VF, or computational simulations, were compared with distributions formed from lifetimes of PS existing at 10,000 simulated commencement timepoints. Results: In all systems, an inspection paradox led towards oversampling of PS with longer lifetimes. In APV computational simulations there was a mean PS lifetime shift of +84.9% (95% CI, ± 0.3%) (p < 0.001 for observed vs overall), in Courtmanche-Nattel simulations of AF +692.9% (95% CI, ±57.7%) (p < 0.001), in optically mapped rat AF +374.6% (95% CI, ± 88.5%) (p = 0.052), in human AF mapped with basket catheters +129.2% (95% CI, ±4.1%) (p < 0.05), human AF-HD grid catheters 150.8% (95% CI, ± 9.0%) (p < 0.001), in optically mapped rat VF +171.3% (95% CI, ±15.6%) (p < 0.001), in human epicardial VF 153.5% (95% CI, ±15.7%) (p < 0.001). Conclusion: Visual inspection of phase movies has the potential to systematically oversample longer lasting PS, due to an inspection paradox. An inspection paradox is minimised by consideration of the overall distribution of PS lifetimes.

Initial perturbation matters: Implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos

Infinitesimal perturbations in various systems showing spatiotemporal chaos (STC) evolve following the power laws of the Kardar-Parisi-Zhang (KPZ) universality class. While universal properties beyond the power-law exponents, such as distributions and correlations and their geometry dependence, are established for random growth and related KPZ systems, the validity of these findings to deterministic chaotic perturbations is unknown. Here, we fill this gap between stochastic KPZ systems and deterministic STC perturbations by conducting extensive simulations of a prototypical STC system, namely, the logistic coupled map lattice. We show that the perturbation interfaces, defined by the logarithm of the modulus of the perturbation vector components, exhibit the universal, geometry-dependent statistical laws of the KPZ class despite the deterministic nature of STC. We demonstrate that KPZ statistics for three established geometries arise for different initial profiles of the perturbation, namely, point (local), uniform, and "pseudo-stationary" initial perturbations, the last being the statistically stationary state of KPZ interfaces given independently of the Lyapunov vector. This geometry dependence lasts until the KPZ correlation length becomes comparable to the system size. Thereafter, perturbation vectors converge to the unique Lyapunov vector, showing characteristic meandering, coalescence, and annihilation of borders of piece-wise regions that remain different from the Lyapunov vector. Our work implies that the KPZ universality for stochastic systems generally characterizes deterministic STC perturbations, providing new insights for STC, such as the universal dependence on initial perturbation and beyond.

Assessing the information content of complex flows

Complex dynamical systems can potentially contain a vast amount of information. Accurately assessing how much of this information must be captured to retain the essential physics is a key step for determining appropriate discretization for numerical simulation or measurement resolution for experiments. Using recent mathematical advances, we define spatiotemporally compact objects that we term dynamical linear neighborhoods (DLNs) that reduce the amount of information needed to capture the local dynamics in a well-defined way. By solving a set-cover problem, we show that we can compress the information in a full dynamical system into a smaller set of optimally influential DLNs. We demonstrate our techniques on experimental data from a laboratory quasi-two-dimensional turbulent flow. Our results have implications both for assessments of the fidelity of simulations or experiments and for the compression of large dynamical data sets.

Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits

In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamical relevance of such solutions by demonstrating that turbulent flows in both experiment and numerics transiently display time-periodic dynamics when they shadow unstable periodic orbits (UPOs). We show that UPOs we computed are also statistically significant, with turbulent flows spending a sizable fraction of the total time near these solutions. As a result, the average rates of energy input and dissipation for the turbulent flow and frequently visited UPOs differ only by a few percent.

Boolean logic by convective obstacle flows

We present a potential new mode of natural computing in which simple, heat-driven fluid flows perform Boolean logic operations. The system comprises a two-dimensional single-phase fluid that is heated from below and cooled from above, with two obstacles placed on the horizontal mid-plane. The obstacles remove all vertical momentum that flows into them. The horizontal momentum extraction of the obstacles is controlled in a binary fashion, and constitutes the 2-bit input. The output of the system is a thresholded measure of the energy extracted by the obstacles. Due to the existence of multiple attractors in the phase space of this system, the input–output relationships are equivalent to those of the OR, XOR or NAND gates, depending on the threshold and obstacle separation. The ability to reproduce these logical operations suggests that convective flows might have the potential to perform more general computations, despite the fact that they do not involve electronics, chemistry or multiple fluid phases.

Thermodynamic analysis of thermal convection based on entropy production

Flow patterns have a tendency to break the symmetry of an initial state of a system and form another spatiotemporal pattern when the system is driven far from equilibrium by temperature difference. For an annular channel, the axially symmetric flow becomes unstable beyond a given temperature difference threshold imposed in the system, leading to rotational oscillating waves. Many researchers have investigated this transition via linear stability analysis using the fundamental conservation equations and the generic model amplitude equation, i.e., the complex Ginzburg-Landau equation. Here, we present a quantitative study conducted of the thermal convection transition using thermodynamic analysis based on the maximum entropy production principle. Our analysis results reveal that the fluid system under nonequilibrium maximizes the entropy production induced by the thermodynamic flux in a direction perpendicular to the temperature difference. Further, we show that the thermodynamic flux as well as the entropy production can uniquely specify the thermodynamic states of the entire fluid system and propose an entropy production selection rule that can be used to specify the thermodynamic state of a nonequilibrium system.

Correlations between the leading Lyapunov vector and pattern defects for chaotic Rayleigh-Bénard convection

We probe the effectiveness of using topological defects to characterize the leading Lyapunov vector for a high-dimensional chaotic convective flow field. This is accomplished using large-scale parallel numerical simulations of Rayleigh-Bénard convection for experimentally accessible conditions. We quantify the statistical correlations between the spatiotemporal dynamics of the leading Lyapunov vector and different measures of the flow field pattern's topology and dynamics. We use a range of pattern diagnostics to describe the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. We use the ideas of precision and recall to build a statistical description of each pattern diagnostic's ability to describe the spatial variation of the leading Lyapunov vector. The precision of a diagnostic indicates the probability that it will locate a region where the Lyapunov vector is larger than a threshold value. The recall of a diagnostic indicates its ability to locate all of the possible spatial regions where the Lyapunov vector is above threshold. By varying the threshold used for the Lyapunov vector magnitude, we generate precision-recall curves which we use to quantify the complex relationship between the pattern diagnostics and the spatiotemporally varying magnitude of the leading Lyapunov vector. We find that pattern diagnostics which include information regarding the flow history outperform pattern diagnostics that do not. In particular, an emerging target defect has the highest precision of all of the pattern diagnostics we have explored.

Convective flow in the presence of a small obstacle: Symmetry breaking, attractors, hysteresis, and information

This work explores the stability and hysteresis effects that occur when a small sink of momentum is introduced into a heat-driven, two-dimensional convective flow. As per standard fluid mechanical intuition, the system minimizes work generation and dissipation when one component of momentum is extracted. However, when the sink absorbs all incoming momentum, the system configures itself such that one of the convection plumes aligns directly with the sink. This state is the most hydrodynamically stable, but it maximizes, rather than minimizes extracted mechanical work. Furthermore, in the case of only vertical momentum extraction, there are two attractors, with different stabilities. Numerical experiments involving slow variations of the horizontal momentum extraction show a clear history dependence. This hysteresis preserves information about the system's past states, and hence represents a primitive memory. The momentum sink can also be used to manipulate the horizontal position of the flow field, with potential applications in microfluidics and laminar convection systems. This simple system exhibits the phenomena of autocatalysis (during the initial growth of the convection plumes), negative feedback (the attractors are either fully or quasistable), memory, and elementary computation.

Morphological Development of Sub-Grain Cellular/Bands Microstructures in Selective Laser Melting

In this paper, single-layer and bulk 316 L selective laser melting (SLM) experiments were conducted, fine submicron-scale geometric symmetrical cellular (hexagonal, pentagonal and square), elongated cellular and bands solidification morphologies were found in the laser-melt top surface. Meanwhile, morphological developed sub-grain patterns with quasi-hexagonal cellular, elongated cellular and bands structures (size ~1 μm) coexisting inside one single macro-solidified grain were also identified. This demonstrated the transitions from quasi-hexagonal-cells to elongated cells/bands, and transitions reverse, occurred in the whole bulk under some circumstances during SLM. Based on the experimental realities, these morphologies are formed by the local convection and Bénard instabilities in front of the solid/liquid interface (so-called mushy zones) affected by intricate temperature and surface tension gradients. Quasi-hexagonal cellular convective fields are then superimposed on macro-grain solidification to form the sub-grain patterns and micro-segregations. This explanation seems reasonable and is unifying as it can be expanded to other eutectic alloys with face center cubic (FCC) prevenient phase prepared by SLM, e.g., the Al-Si and Co-Cr-Mo systems.

Extinction dynamics of spiral defect chaos

Spatially extended excitable systems can exhibit spiral defect chaos (SDC) during which spiral waves continuously form and disappear. To address how this dynamical state terminates using simulations can be computationally challenging, especially for large systems. To circumvent this limitation, we treat the number of spiral waves as a stochastic population with a corresponding birth-death equation and use techniques from statistical physics to determine the mean episode duration of SDC. Motivated by cardiac fibrillation, during which the heart's electrical activity becomes disorganized and shows fragmenting spiral waves, we use generic models of cardiac electrophysiology. We show that the duration can be computed in minimal computational time and that it depends exponentially on domain size. Therefore, the approach can result in efficient and accurate predictions of mean episode duration which may be extended to more complex geometries and models.

New perspectives for the prediction and statistical quantification of extreme events in high-dimensional dynamical systems

We discuss extreme events as random occurrences of strongly transient dynamics that lead to nonlinear energy transfers within a chaotic attractor. These transient events are the result of finite-time instabilities and therefore are inherently connected with both statistical and dynamical properties of the system. We consider two classes of problems related to extreme events and nonlinear energy transfers, namely (i) the derivation of precursors for the short-term prediction of extreme events, and (ii) the efficient sampling of random realizations for the fastest convergence of the probability density function in the tail region. We summarize recent methods on these problems that rely on the simultaneous consideration of the statistical and dynamical characteristics of the system. This is achieved by combining available data, in the form of second-order statistics, with dynamical equations that provide information for the transient events that lead to extreme responses. We present these methods through two high-dimensional, prototype systems that exhibit strongly chaotic dynamics and extreme responses due to transient instabilities, the Kolmogorov flow and unidirectional nonlinear water waves.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Chaotic Rayleigh-Bénard convection with finite sidewalls

We explore the role of finite sidewalls on chaotic Rayleigh-Bénard convection. We use large-scale parallel spectral-element numerical simulations for the precise conditions of experiment for cylindrical convection domains. We solve the Boussinesq equations for thermal convection and the conjugate heat transfer problem for the energy transfer at the solid sidewalls of the cylindrical domain. The solid sidewall of the convection domain has finite values of thickness, thermal conductivity, and thermal diffusivity. We compute the Lyapunov vectors and exponents for the entire fluid-solid coupled problem. We quantify the chaotic dynamics of convection over a range of thermal sidewall boundary conditions. We find that the dynamics become less chaotic as the thermal conductivity of the sidewalls increases as measured by the value of the fractal dimension of the dynamics. The thermal conductivity of the sidewall is a stabilizing influence; the heat transfer between the fluid and solid regions is always in the direction to reduce the fluid motion near the sidewalls. Although the heat interaction for strongly conducting sidewalls is only about 1% of the heat transfer through the fluid layer, it is sufficient to reduce the fractal dimension of the dynamics by approximately 25% in our computations.

Spatiotemporal dynamics of the covariant Lyapunov vectors of chaotic convection

We explore the spatiotemporal dynamics of the spectrum of covariant Lyapunov vectors for chaotic Rayleigh-Bénard convection. We use the inverse participation ratio to quantify the amount of spatial localization of the covariant Lyapunov vectors. The covariant Lyapunov vectors are found to be spatially localized at times when the instantaneous covariant Lyapunov exponents are large. The spatial localization of the Lyapunov vectors often occurs near defect structures in the fluid flow field. There is an overall trend of decreasing spatial localization of the Lyapunov vectors with increasing index of the vector. The spatial localization of the covariant Lyapunov vectors with positive Lyapunov exponents decreases an order of magnitude faster with increasing vector index than all of the remaining vectors that we have computed. We find that a weighted covariant Lyapunov vector is useful for the visualization and interpretation of the significant connections between the Lyapunov vectors and the flow field patterns.

Finding equilibrium in the spatiotemporal chaos of the complex Ginzburg-Landau equation

Equilibrium statistical mechanics allows the prediction of collective behaviors of large numbers of interacting objects from just a few system-wide properties; however, a similar theory does not exist for far-from-equilibrium systems exhibiting complex spatial and temporal behavior. We propose a method for predicting behaviors in a broad class of such systems and apply these ideas to an archetypal example, the spatiotemporal chaotic 1D complex Ginzburg-Landau equation in the defect chaos regime. Building on the ideas of Ruelle and of Cross and Hohenberg that a spatiotemporal chaotic system can be considered a collection of weakly interacting dynamical units of a characteristic size, the chaotic length scale, we identify underlying, mesoscale, chaotic units and effective interaction potentials between them. We find that the resulting equilibrium Takahashi model accurately predicts distributions of particle numbers. These results suggest the intriguing possibility that a class of far-from-equilibrium systems may be well described at coarse-grained scales by the well-established theory of equilibrium statistical mechanics.

Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems

Drawing upon the bursting mechanism in slow-fast systems, we propose indicators for the prediction of such rare extreme events which do not require a priori known slow and fast coordinates. The indicators are associated with functionals defined in terms of optimally time-dependent (OTD) modes. One such functional has the form of the largest eigenvalue of the symmetric part of the linearized dynamics reduced to these modes. In contrast to other choices of subspaces, the proposed modes are flow invariant and therefore a projection onto them is dynamically meaningful. We illustrate the application of these indicators on three examples: a prototype low-dimensional model, a body-forced turbulent fluid flow, and a unidirectional model of nonlinear water waves. We use Bayesian statistics to quantify the predictive power of the proposed indicators.

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.

A minimization principle for the description of modes associated with finite-time instabilities

We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime, they can play a crucial role either by altering the system dynamics through the activation of other instabilities or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighbourhood of the system state. This minimization formulation results in differential equations that evolve a time-dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: (i) linear systems, including the advection-diffusion operator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squire operator, and (ii) nonlinear problems, including a low-dimensional system with transient instabilities and the vertical jet in cross-flow. We demonstrate that the time-dependent subspace captures the strongly transient non-normal energy growth (in the short-time regime), while for longer times the modes capture the expected asymptotic behaviour.

Chaos, patterns, coherent structures, and turbulence: Reflections on nonlinear science

The paradigms of nonlinear science were succinctly articulated over 25 years ago as deterministic chaos, pattern formation, coherent structures, and adaptation/evolution/learning. For chaos, the main unifying concept was universal routes to chaos in general nonlinear dynamical systems, built upon a framework of bifurcation theory. Pattern formation focused on spatially extended nonlinear systems, taking advantage of symmetry properties to develop highly quantitative amplitude equations of the Ginzburg-Landau type to describe early nonlinear phenomena in the vicinity of critical points. Solitons, mathematically precise localized nonlinear wave states, were generalized to a larger and less precise class of coherent structures such as, for example, concentrated regions of vorticity from laboratory wake flows to the Jovian Great Red Spot. The combination of these three ideas was hoped to provide the tools and concepts for the understanding and characterization of the strongly nonlinear problem of fluid turbulence. Although this early promise has been largely unfulfilled, steady progress has been made using the approaches of nonlinear science. I provide a series of examples of bifurcations and chaos, of one-dimensional and two-dimensional pattern formation, and of turbulence to illustrate both the progress and limitations of the nonlinear science approach. As experimental and computational methods continue to improve, the promise of nonlinear science to elucidate fluid turbulence continues to advance in a steady manner, indicative of the grand challenge nature of strongly nonlinear multi-scale dynamical systems.

Pattern-fluid interpretation of chemical turbulence

The spontaneous formation of heterogeneous patterns is a hallmark of many nonlinear systems, from biological tissue to evolutionary population dynamics. The standard model for pattern formation in general, and for Turing patterns in chemical reaction-diffusion systems in particular, are deterministic nonlinear partial differential equations where an unstable homogeneous solution gives way to a stable heterogeneous pattern. However, these models fail to fully explain the experimental observation of turbulent patterns with spatio-temporal disorder in chemical systems. Here we introduce a pattern-fluid model as a general concept where turbulence is interpreted as a weakly interacting ensemble obtained by random superposition of stationary solutions to the underlying reaction-diffusion system. The transition from turbulent to stationary patterns is then interpreted as a condensation phenomenon, where the nonlinearity forces one single mode to dominate the ensemble. This model leads to better reproduction of the experimental concentration profiles for the "stationary phases" and reproduces the turbulent chemical patterns observed by Q. Ouyang and H. L. Swinney [Chaos 1, 411 (1991)].

A PREDATOR–PREY INTERACTION MODEL WITH SELF- AND CROSS-DIFFUSION IN AQUATIC SYSTEMS

In this paper, the complex dynamics of a spatial aquatic system in the presence of self- and cross-diffusion are investigated. Criteria for local stability, instability and global stability are obtained. The effect of critical wavelength which can drive a system to instability is investigated. We noticed that cross-diffusion coefficient can be quite significant, even for small values of off-diagonal terms in the diffusion matrix. With the help of numerical simulation, we observed the Turing patterns (spots, strips, spot-strips mixture), regular spiral patterns and irregular patchy structures. The beauty and complexity of the Turing patterns are attributed to a large variety of symmetry properties realized by different values of predator's immunity, rate of fish predation and half saturation constant of predator population.

Front propagation in a chaotic flow field

We investigate numerically the dynamics of a propagating front in the presence of a spatiotemporally chaotic flow field. The flow field is the three-dimensional time-dependent state of spiral defect chaos generated by Rayleigh-Bénard convection in a spatially extended domain. Using large-scale parallel numerical simulations, we simultaneously solve the Boussinesq equations and a reaction-advection-diffusion equation with a Fischer-Kolmogorov-Petrovskii-Piskunov reaction for the transport of the scalar species in a large-aspect-ratio cylindrical domain for experimentally accessible conditions. We explore the front dynamics and geometry in the low-Damköhler-number regime, where the effect of the flow field is significant. Our results show that the chaotic flow field enhances the front propagation when compared with a purely cellular flow field. We quantify this enhancement by computing the spreading rate of the reaction products for a range of parameters. We use our results to quantify the complexity of the three-dimensional front geometry for a range of chaotic flow conditions.

Quasiperiodicity route to spatiotemporal chaos in one-dimensional pattern-forming systems

We propose a route to spatiotemporal chaos for one-dimensional stationary patterns, which is a natural extension of the quasiperiodicity route for low-dimensional chaos to extended systems. This route is studied through a universal model of pattern formation. The model exhibits a scenario where stationary patterns become spatiotemporally chaotic through two successive bifurcations. First, the pattern undergoes a subcritical Andronov-Hopf bifurcation leading to an oscillatory pattern. Subsequently, a secondary bifurcation gives rise to an oscillation with an incommensurable frequency with respect to the former one. This last bifurcation is responsible for the spatiotemporally chaotic behavior. The Lyapunov spectrum enables us to identify the complex behavior observed as spatiotemporal chaos, and also from the larger Lyapunov exponents characterize the above instabilities.

Microscopic insight into the bilateral formation of carbon spirals from a symmetric iron core

Mirrored carbon-spirals have been produced from pressured ferrocene via the bilateral extrusion of the spiral pairs from an iron core. A parametric plot of the surface geometry displays the fractal growth of the conical helix made with the logarithmic spiral. Electron microscopy studies show the core is a crystalline cementite which grows and transforms its shape from spherical to biconical as it extrudes two spiralling carbon arms. In a cross section along the arms we observe graphitic flakes arranged in a herringbone structure, normal to which defects propagate. Local-wave-pattern analysis reveals nanoscale defect patterns of two-fold symmetry around the core. The data suggest that the bilateral growth originates from a globular cementite crystal with molten surfaces and the nano-defects shape emerging hexagonal carbon into a fractal structure. Understanding and knowledge obtained provide a basis for the controlled production of advanced carbon materials with designed geometries.

On the existence and multiplicity of one-dimensional solid particle attractors in time-dependent Rayleigh-Bénard convection

For the first time evidence is provided that one-dimensional objects formed by the accumulation of tracer particles can emerge in flows of thermogravitational nature (in the region of the space of parameters, in which the so-called OS (oscillatory solution) flow of the Busse balloon represents the dominant secondary mode of convection). Such structures appear as seemingly rigid filaments, rotating without changing their shape. The most interesting (heretofore unseen) feature of such a class of physical attractors is their variety. Indeed, distinct shapes are found for a fixed value of the Rayleigh number depending on parameters accounting for particle inertia and viscous drag. The fascinating "sea" of existing potential paths, their multiplicity and tortuosity are explained according to the granularity of the loci in the physical space where conditions for phase locking between the traveling thermofluid-dynamic disturbance and the "turnover time" of particles in the basic toroidal flow are satisfied. It is shown, in particular, how the observed wealth of geometric objects and related topological features can be linked to a general overarching attractor representing an intrinsic (particle-independent) property of the base velocity field.

Length scale of a chaotic element in Rayleigh-Bénard convection

We describe an approach to quantify the length scale of a chaotic element of a Rayleigh-Bénard convection layer exhibiting spatiotemporal chaos. The length scale of a chaotic element is determined by simultaneously evolving the dynamics of two convection layers with a unidirectional coupling that involves only the time-varying values of the fluid velocity and temperature on the lateral boundaries of the domain. In our results we numerically simulate the full Boussinesq equations for the precise conditions of experiment. By varying the size of the boundary used for the coupling we identify a length scale that describes the size of a chaotic element. The length scale of the chaotic element is of the same order of magnitude, and exhibits similar trends, as the natural chaotic length scale that is based upon the fractal dimension.

Quantifying spatiotemporal chaos in Rayleigh-Bénard convection

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading-order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading-order Lyapunov exponent, and we quantify the details of their response to the dynamics of defects. The leading-order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary-dominated dynamics to bulk-dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.

Covariant Lyapunov analysis of chaotic Kolmogorov flows

Hyperbolicity is an important concept in dynamical system theory; however, we know little about the hyperbolicity of concrete physical systems including fluid motions governed by the Navier-Stokes equations. Here, we study numerically the hyperbolicity of the Navier-Stokes equation on a two-dimensional torus (Kolmogorov flows) using the method of covariant Lyapunov vectors developed by Ginelli et al. [Phys. Rev. Lett. 99, 130601 (2007)]. We calculate the angle between the local stable and unstable manifolds along an orbit of chaotic solution to evaluate the hyperbolicity. We find that the attractor of chaotic Kolmogorov flows is hyperbolic at small Reynolds numbers, but that smaller angles between the local stable and unstable manifolds are observed at larger Reynolds numbers, and the attractor appears to be nonhyperbolic at a certain Reynolds numbers. Also, we observed some relations between these hyperbolic properties and physical properties such as time correlation of the vorticity and the energy dissipation rate.

Extensive and subextensive chaos in globally coupled dynamical systems

Using a combination of analytical and numerical techniques, we show that chaos in globally coupled identical dynamical systems, whether dissipative or Hamiltonian, is both extensive and subextensive: their spectrum of Lyapunov exponents is asymptotically flat (thus extensive) at the value λ(0) given by a single unit forced by the mean field, but sandwiched between subextensive bands containing typically O(logN) exponents whose values vary as λ≃λ(∞)+c/logN with λ(∞)≠λ(0).

Modal spectra extracted from nonequilibrium fluid patterns in laboratory experiments on Rayleigh-Bénard convection

We describe a method to extract from experimental data the important dynamical modes in spatiotemporal patterns in a system driven out of thermodynamic equilibrium. Using a novel optical technique for controlling fluid flow, we create an experimental ensemble of Rayleigh-Bénard convection patterns with nearby initial conditions close to the onset of secondary instability. An analysis of the ensemble evolution reveals the spatial structure of the dominant modes of the system as well as the corresponding growth rates. The extracted modes are related to localized versions of instabilities found in the ideal unbounded system. The approach may prove useful in describing instability in experimental systems as a step toward prediction and control.

Extensive scaling from computational homology and Karhunen-Loève decomposition analysis of Rayleigh-Bénard convection experiments

Spatiotemporally chaotic dynamics in laboratory experiments on convection are characterized using a new dimension, D(CH), determined from computational homology. Over a large range of system sizes, D(CH) scales in the same manner as D(KLD), a dimension determined from experimental data using Karhuenen-Loéve decomposition. Moreover, finite-size effects (the presence of boundaries in the experiment) lead to deviations from scaling that are similar for both D(CH) and D(KLD). In the absence of symmetry, D(CH) can be determined more rapidly than D(KLD).

Extensive chaos in the Lorenz-96 model

We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and the deviation from extensivity decreases nonmonotonically with increasing system size. The length scale describing the deviations from extensivity is consistent with the natural chaotic length scale in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. We find that each wavelength of the deviation from extensive chaos contains on the order of two chaotic degrees of freedom. As the forcing is increased, at constant system size, the dimension density grows monotonically and saturates at a value less than unity. We use this to quantify the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns.

Structure of characteristic Lyapunov vectors in anharmonic Hamiltonian lattices

In this work we perform a detailed study of the scaling properties of Lyapunov vectors (LVs) for two different one-dimensional Hamiltonian lattices: the Fermi-Pasta-Ulam and Φ^{4} models. In this case, characteristic (also called covariant) LVs exhibit qualitative similarities with those of dissipative lattices but the scaling exponents are different and seemingly nonuniversal. In contrast, backward LVs (obtained via Gram-Schmidt orthonormalizations) present approximately the same scaling exponent in all cases, suggesting it is an artificial exponent produced by the imposed orthogonality of these vectors. We are able to compute characteristic LVs in large systems thanks to a "bit reversible" algorithm, which completely obviates computer memory limitations.

Diagnosis of spatiotemporal chaos in wave envelopes of a nematic electroconvection pattern

In this paper we report and analyze complex spatiotemporal dynamics recorded in electroconvection in the nematic liquid crystal I52, driven by an ac voltage slightly above the onset value. The instability mechanism creating the pattern is an oscillatory (Hopf) instability, giving rise to two pairs of counterpropagating rolls traveling in oblique directions relative to the unperturbed director axis. If a system of nonlinear partial differential equations shows the same set of unstable modes, the pattern above the onset is represented in a weakly nonlinear analysis as a superposition of the traveling rolls in terms of wave envelopes varying slowly in space and time. Motivated by this procedure, we extract slowly varying envelopes from the space-time data of the pattern, using a four-wave demodulation based on Fourier analysis. In order to characterize the spatiotemporal dynamics, we apply a variety of diagnostic methods to the envelopes, including the calculation of mean intensities and correlation lengths, global and local Karhunen-Loève decompositions in Fourier space and physical space, the location of holes, the identification of coherent vertical structures, and estimates of Lyapunov exponents. The results of this analysis provide strong evidence that our pattern exhibits extensive spatiotemporal chaos. One of its main characteristics is the presence of coherent structures of low and high intensities extended in the vertical (parallel to the director) direction.

State and parameter estimation of spatiotemporally chaotic systems illustrated by an application to Rayleigh-Bénard convection

Data assimilation refers to the process of estimating a system's state from a time series of measurements (which may be noisy or incomplete) in conjunction with a model for the system's time evolution. Here we demonstrate the applicability of a recently developed data assimilation method, the local ensemble transform Kalman filter, to nonlinear, high-dimensional, spatiotemporally chaotic flows in Rayleigh-Bénard convection experiments. Using this technique we are able to extract the full temperature and velocity fields from a time series of shadowgraph measurements. In addition, we describe extensions of the algorithm for estimating model parameters. Our results suggest the potential usefulness of our data assimilation technique to a broad class of experimental situations exhibiting spatiotemporal chaos.

Parabolic resonance: a route to hamiltonian spatiotemporal chaos

We show that initial data near an unperturbed stable plane wave can evolve into a regime of spatiotemporal chaos in the slightly forced conservative periodic one-dimensional nonlinear Schrödinger equation. Statistical measures are employed to demonstrate that this spatiotemporal chaos is intermittent: there are windows in time for which the solution gains spatial coherence. The parameters and initial profiles that lead to such intermittency are predicted by utilizing a novel geometrical description of the integrable unforced equation.

Self-organization of topological defects due to applied constraints

While topological defects are essential to our understanding of self-organizing periodic systems, little is known about how these systems respond when their defects are subjected to geometrical constraints. In an experiment on spatially modulated thermal convection patterns, we observe that applied geometrical constraints bind topological defects into robust self-localized structures that evolve through aggregation, annihilation, and self-replication. We demonstrate that this unexpected cooperative response to the modulation is a natural consequence of three generic elements: phase locking, symmetry breaking, and spatial resonance. Our work provides new insights into the interplay between order, chaos, and control in self-organizing systems.

Structure of characteristic Lyapunov vectors in spatiotemporal chaos

We study Lyapunov vectors (LVs) corresponding to the largest Lyapunov exponents in systems with spatiotemporal chaos. We focus on characteristic LVs and compare the results with backward LVs obtained via successive Gram-Schmidt orthonormalizations. Systems of a very different nature such as coupled-map lattices and the (continuous-time) Lorenz '96 model exhibit the same features in quantitative and qualitative terms. Additionally, we propose a minimal stochastic model that reproduces the results for chaotic systems. Our work supports the claims about universality of our earlier results [I. G. Szendro, Phys. Rev. E 76, 025202(R) (2007)] for a specific coupled-map lattice.

Curvature fields, topology, and the dynamics of spatiotemporal chaos

The curvature field is measured from tracer-particle trajectories in a two-dimensional fluid flow that exhibits spatiotemporal chaos and is used to extract the hyperbolic and elliptic points of the flow. These special points are pinned to the forcing when the driving is weak, but wander over the domain and interact in pairs at stronger driving, changing the local topology of the flow. Their behavior reveals a two-stage transition to spatiotemporal chaos: a gradual loss of spatial and temporal order followed by an abrupt onset of topological changes.

Characterizing dynamics with covariant Lyapunov vectors

A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows us to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents.

Spatiotemporal structure of Lyapunov vectors in chaotic coupled-map lattices

The spatiotemporal dynamics of Lyapunov vectors (LVs) in spatially extended chaotic systems is studied by means of coupled-map lattices. We determine intrinsic length scales and spatiotemporal correlations of LVs corresponding to the leading unstable directions by translating the problem to the language of scale-invariant growing surfaces. We find that the so-called characteristic LVs exhibit spatial localization, strong clustering around given spatiotemporal loci, and remarkable dynamic scaling properties of the corresponding surfaces. In contrast, the commonly used backward LVs (obtained through Gram-Schmidt orthogonalization) spread all over the system and do not exhibit dynamic scaling due to artifacts in the dynamical correlations by construction.

A dynamically changing intracellular water network serves as a universal regulator of the cell: the water-governed cycle

The functioning of enzymes and protein folding are well known to be assisted by the surrounding chaperoning water molecules, which are connected to the protein via non-covalent, dynamically changing chemical bonds. A molecular intracellular network of weak non-covalent connections may be presumed to exist in living cells. The roles of such non-covalent networks are examined in terms of a molecular model which postulates a universal enzyme and biochemical mechanism regulating the maintenance of chemical stability in living cells.

Extensive chaos in Rayleigh-Bénard convection

Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size.

Lyapunov exponents for small aspect ratio Rayleigh-Bénard convection

Leading order Lyapunov exponents and their corresponding eigenvectors have been computed numerically for small aspect ratio, three-dimensional Rayleigh-Benard convection cells with no-slip boundary conditions. The parameters are the same as those used by Ahlers and Behringer [Phys. Rev. Lett. 40, 712 (1978)] and Gollub and Benson [J. Fluid Mech. 100, 449 (1980)] in their work on a periodic time dependence in Rayleigh-Benard convection cells. Our work confirms that the dynamics in these cells truly are chaotic as defined by a positive Lyapunov exponent. The time evolution of the leading order Lyapunov eigenvector in the chaotic regime will also be discussed. In addition we study the contributions to the leading order Lyapunov exponent for both time periodic and aperiodic states and find that while repeated dynamical events such as dislocation creation/annihilation and roll compression do contribute to the short time Lyapunov exponent dynamics, they do not contribute to the long time Lyapunov exponent. We find instead that nonrepeated events provide the most significant contribution to the long time leading order Lyapunov exponent.