Characterizing dynamics with covariant Lyapunov vectors

Exploring the role of diffusive coupling in spatiotemporal chaos

We explore the chaotic dynamics of a large one-dimensional lattice of coupled maps with diffusive coupling of varying strength using the covariant Lyapunov vectors (CLVs). Using a lattice of diffusively coupled quadratic maps, we quantify the growth of spatial structures in the chaotic dynamics as the strength of diffusion is increased. When the diffusion strength is increased from zero, we find that the leading Lyapunov exponent decreases rapidly from a positive value to zero to yield a small window of periodic dynamics which is then followed by chaotic dynamics. For values of the diffusion strength beyond the window of periodic dynamics, the leading Lyapunov exponent does not vary significantly with the strength of diffusion with the exception of a small variation for the largest diffusion strengths we explore. The Lyapunov spectrum and fractal dimension are described analytically as a function of the diffusion strength using the eigenvalues of the coupling operator. The spatial features of the CLVs are quantified and compared with the eigenvectors of the coupling operator. The chaotic dynamics are composed entirely of physical modes for all of the conditions we explore. The leading CLV is highly localized and localization decreases with increasing strength of the spatial coupling. The violation of the dominance of Oseledets splitting indicates that the entanglement of pairs of CLVs becomes more significant between neighboring CLVs as the strength of diffusion is increased.

On non-trivial hyperbolic sets and their bifurcations in families of diffeomorphisms of a two-dimensional torus

We propose a simple model-two-parameter family of diffeomorphisms of a two-dimensional torus. Combining analytical and numerical methods, we find regions in the parameter plane such that each diffeomorphism of the family is hyperbolic and describe the structure of the corresponding hyperbolic sets. We also study bifurcations on the boundaries of these regions, which lead to the change of hyperbolicity type (from Anosov diffeomorphisms to DA-diffeomorphisms).

Lyapunov vectors and excited energy states of the directed polymer in random media

The scaling behavior of the excited energy states of the directed polymer in random media is analyzed numerically. We find that the spatial correlations of polymer energies scale as ∼k^{-δ} for small enough wave numbers k with a nontrivial exponent δ≈1.3. The equivalence between the stochastic-field equation that describes the partition function of the directed polymer and that governing the time evolution of infinitesimal perturbations in space-time chaos is exploited to connect this exponent δ with the spatial correlations of the Lyapunov vectors reported in the literature. The relevance of our results for other problems involving optimization in random systems is discussed.

Characterizing Small-Scale Dynamics of Navier-Stokes Turbulence with Transverse Lyapunov Exponents: A Data Assimilation Approach

Data assimilation (DA) of turbulence, which involves reconstructing small-scale turbulent structures based on observational data from large-scale ones, is crucial not only for practical forecasting but also for gaining a deeper understanding of turbulent dynamics. We propose a theoretical framework for DA of turbulence based on the transverse Lyapunov exponents (TLEs) in synchronization theory. Through stability analysis using TLEs, we identify a critical length scale as a key condition for DA; turbulent dynamics smaller than this scale are synchronized with larger-scale turbulent dynamics. Furthermore, considering recent findings for the maximal Lyapunov exponent and its relation with the TLEs, we clarify the Reynolds number dependence of the critical length scale.

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.

Stability analysis of chaotic systems from data

The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point and compute the properties of the tangent space (i.e. the Jacobian). The main goal of this paper is to propose a method that infers the Jacobian, thus, the stability properties, from observables (data). First, we propose the echo state network (ESN) with the Recycle validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the echo state network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution and its tangent space with negligible numerical errors. In detail, we compute from data only (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (iii) the Lyapunov spectrum; (iv) the finite-time Lyapunov exponents; (v) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11071-023-08285-1.

Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems

Dynamical vectors characterizing instability and applicable as ensemble perturbations for prediction with geophysical fluid dynamical models are analysed. The relationships between covariant Lyapunov vectors (CLVs), orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors and finite-time normal modes (FTNMs) are examined for periodic and aperiodic systems. In the phase-space of FTNM coefficients, SVs are shown to equate with unit norm FTNMs at critical times. In the long-time limit, when SVs approach OLVs, the Oseledec theorem and the relationships between OLVs and CLVs are used to connect CLVs to FTNMs in this phase-space. The covariant properties of both the CLVs, and the FTNMs, together with their phase-space independence, and the norm independence of global Lyapunov exponents and FTNM growth rates, are used to establish their asymptotic convergence. Conditions on the dynamical systems for the validity of these results, particularly ergodicity, boundedness and non-singular FTNM characteristic matrix and propagator, are documented. The findings are deduced for systems with nondegenerate OLVs, and, as well, with degenerate Lyapunov spectrum as is the rule in the presence of waves such as Rossby waves. Efficient numerical methods for the calculation of leading CLVs are proposed. Norm independent finite-time versions of the Kolmogorov-Sinai entropy production and Kaplan-Yorke dimension are presented.

Finite-time Lyapunov fluctuations and the upper bound of classical and quantum out-of-time-ordered expansion rate exponents

This Letter demonstrates for chaotic maps [logistic, classical, and quantum standard maps (SMs)] that the exponential growth rate (Λ) of the out-of-time-ordered four-point correlator is equal to the classical Lyapunov exponent (λ) plus fluctuations (Δ^{(fluc)}) of the one-step finite-time Lyapunov exponents (FTLEs). Jensen's inequality provides the upper bound λ≤Λ for the considered systems. Equality is restored with Λ=λ+Δ^{(fluc)}, where Δ^{(fluc)} is quantified by k-higher-order cumulants of the (covariant) FTLEs. Exact expressions for Λ are derived and numerical results using k=20 furnish Δ^{(fluc)}∼ln(sqrt[2]) for all maps (large kicking intensities in the SMs).

Guidelines for data-driven approaches to study transitions in multiscale systems: The case of Lyapunov vectors

This study investigates the use of covariant Lyapunov vectors and their respective angles for detecting transitions between metastable states in dynamical systems, as recently discussed in several atmospheric sciences applications. In a first step, the needed underlying dynamical models are derived from data using a non-parametric model-based clustering framework. The covariant Lyapunov vectors are then approximated based on these data-driven models. The data-based numerical approach is tested using three well-understood example systems with increasing dynamical complexity, identifying properties that allow for a successful application of the method: in particular, the method is identified to require a clear multiple time scale structure with fast transitions between slow subsystems. The latter slow dynamics should be dynamically characterized by invariant neutral directions of the linear approximation model.

Density matrix formulation of dynamical systems

Physical systems that are dissipating, mixing, and developing turbulence also irreversibly transport statistical density. However, predicting the evolution of density from atomic and molecular scale dynamics is challenging for nonsteady, open, and driven nonequilibrium processes. Here, we establish a theory to address this challenge for classical dynamical systems that is analogous to the density matrix formulation of quantum mechanics. We show that a classical density matrix is similar to the phase-space metric and evolves in time according to generalizations of Liouville's theorem and Liouville's equation for non-Hamiltonian systems. The traditional Liouvillian forms are recovered in the absence of dissipation or driving by imposing trace preservation or by considering Hamiltonian dynamics. Local measures of dynamical instability and chaos are embedded in classical commutators and anticommutators and directly related to Poisson brackets when the dynamics are Hamiltonian. Because the classical density matrix is built from the Lyapunov vectors that underlie classical chaos, it offers an alternative computationally tractable basis for the statistical mechanics of nonequilibrium processes that applies to systems that are driven, transient, dissipative, regular, and chaotic.

Phase and frequency linear response theory for hyperbolic chaotic oscillators

We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture, stating the existence of a perturbed trajectory shadowing every unperturbed trajectory on the system attractor for any small enough perturbation of arbitrary duration and a corresponding unique time isomorphism, which we identify as phase such that phase shifts between the unperturbed trajectory and its perturbed shadow are well defined. The phase sensitivity function is the solution of an adjoint linear equation and can be used to estimate the average change of phase velocity to small time dependent or independent perturbations. These changes in frequency are experimentally accessible, giving a convenient way to define and measure phase response curves for chaotic oscillators. The shadowing trajectory and the phase can be constructed explicitly in the tangent space of an unperturbed trajectory using co-variant Lyapunov vectors. It can also be used to identify the limits of the regime of linear response.

Estimating covariant Lyapunov vectors from data

Covariant Lyapunov vectors characterize the directions along which perturbations in dynamical systems grow. They have also been studied as predictors of critical transitions and extreme events. For many applications, it is necessary to estimate these vectors from data since model equations are unknown for many interesting phenomena. We propose an approach for estimating covariant Lyapunov vectors based on data records without knowing the underlying equations of the system. In contrast to previous approaches, our approach can be applied to high-dimensional datasets. We demonstrate that this purely data-driven approach can accurately estimate covariant Lyapunov vectors from data records generated by several low- and high-dimensional dynamical systems. The highest dimension of a time series from which covariant Lyapunov vectors are estimated in this contribution is 128.

Coherent oscillations in balanced neural networks driven by endogenous fluctuations

We present a detailed analysis of the dynamical regimes observed in a balanced network of identical quadratic integrate-and-fire neurons with sparse connectivity for homogeneous and heterogeneous in-degree distributions. Depending on the parameter values, either an asynchronous regime or periodic oscillations spontaneously emerge. Numerical simulations are compared with a mean-field model based on a self-consistent Fokker-Planck equation (FPE). The FPE reproduces quite well the asynchronous dynamics in the homogeneous case by either assuming a Poissonian or renewal distribution for the incoming spike trains. An exact self-consistent solution for the mean firing rate obtained in the limit of infinite in-degree allows identifying balanced regimes that can be either mean- or fluctuation-driven. A low-dimensional reduction of the FPE in terms of circular cumulants is also considered. Two cumulants suffice to reproduce the transition scenario observed in the network. The emergence of periodic collective oscillations is well captured both in the homogeneous and heterogeneous setups by the mean-field models upon tuning either the connectivity or the input DC current. In the heterogeneous situation, we analyze also the role of structural heterogeneity.

Systematic calculation of finite-time mixed singular vectors and characterization of error growth for persistent coherent atmospheric disturbances over Eurasia

Singular vectors (SVs) have long been employed in the initialization of ensemble numerical weather prediction (NWP) in order to capture the structural organization and growth rates of those perturbations or "errors" associated with initial condition errors and instability processes of the large scale flow. Due to their (super) exponential growth rates and spatial scales, initial SVs are typically combined empirically with evolved SVs in order to generate forecast perturbations whose structures and growth rates are tuned for specified lead-times. Here, we present a systematic approach to generating finite time or "mixed" SVs (MSVs) based on a method for the calculation of covariant Lyapunov vectors and appropriate choices of the matrix cocycle. We first derive a data-driven reduced-order model to characterize persistent geopotential height anomalies over Europe and Western Asia (Eurasia) over the period 1979-present from the National Centers for Environmental Prediction v1 reanalysis. We then characterize and compare the MSVs and SVs of each persistent state over Eurasia for particular lead-times from a day to over a week. Finally, we compare the spatiotemporal properties of SVs and MSVs in an examination of the dynamics of the 2010 Russian heatwave. We show that MSVs provide a systematic approach to generate initial forecast perturbations projected onto relevant expanding directions in phase space for typical NWP forecast lead-times.

Dynamical system analysis of a data-driven model constructed by reservoir computing

This study evaluates data-driven models from a dynamical system perspective, such as unstable fixed points, periodic orbits, chaotic saddle, Lyapunov exponents, manifold structures, and statistical values. We find that these dynamical characteristics can be reconstructed much more precisely by a data-driven model than by computing directly from training data. With this idea, we predict the laminar lasting time distribution of a particular macroscopic variable of chaotic fluid flow, which cannot be calculated from a direct numerical simulation of the Navier-Stokes equation because of its high computational cost.

Sensitivity of long periodic orbits of chaotic systems

The properties of long, numerically determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed. The expectation of period averages, Floquet exponents, and sensitivities over such set is then obtained. It is shown that all these quantities converge to a limiting value as the orbit period is increased. However, while period averages and Floquet exponents appear to converge to analogous quantities computed from chaotic trajectories, the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state, similar to observations made with other shadowing algorithms.

Route to hyperbolic hyperchaos in a nonautonomous time-delay system

We consider a self-oscillator whose excitation parameter is varied. The frequency of the variation is much smaller than the natural frequency of the oscillator so that oscillations in the system are periodically excited and decayed. Also, a time delay is added such that when the oscillations start to grow at a new excitation stage, they are influenced via the delay line by the oscillations at the penultimate excitation stage. Due to nonlinearity, the seeding from the past arrives with a doubled phase so that the oscillation phase changes from stage to stage according to the chaotic Bernoulli-type map. As a result, the system operates as two coupled hyperbolic chaotic subsystems. Varying the relation between the delay time and the excitation period, we found a coupling strength between these subsystems as well as intensity of the phase doubling mechanism responsible for the hyperbolicity. Due to this, a transition from non-hyperbolic to hyperbolic hyperchaos occurs. The following steps of the transition scenario are revealed and analyzed: (a) an intermittency as an alternation of long staying near a fixed point at the origin and short chaotic bursts; (b) chaotic oscillations with frequent visits to the fixed point; (c) plain hyperchaos without hyperbolicity after termination visiting the fixed point; and (d) transformation of hyperchaos to the hyperbolic form.

Predicting regime changes and durations in Lorenz's atmospheric convection model

We show that a characteristic alignment between Lyapunov vectors can be used to predict regime changes as well as regime duration in the classical Lorenz model of atmospheric convection. By combining Lyapunov vector alignment with maxima in the local expansion of bred vectors, we obtain an effective and competitive method to significantly decrease errors in the prediction of regime durations.

Dynamics and computation in mixed networks containing neurons that accelerate towards spiking

Networks in the brain consist of different types of neurons. Here we investigate the influence of neuron diversity on the dynamics, phase space structure, and computational capabilities of spiking neural networks. We find that already a single neuron of a different type can qualitatively change the network dynamics and that mixed networks may combine the computational capabilities of ones with a single-neuron type. We study inhibitory networks of concave leaky (LIF) and convex "antileaky" (XIF) integrate-and-fire neurons that generalize irregularly spiking nonchaotic LIF neuron networks. Endowed with simple conductance-based synapses for XIF neurons, our networks can generate a balanced state of irregular asynchronous spiking as well. We determine the voltage probability distributions and self-consistent firing rates assuming Poisson input with finite-size spike impacts. Further, we compute the full spectrum of Lyapunov exponents (LEs) and the covariant Lyapunov vectors (CLVs) specifying the corresponding perturbation directions. We find that there is approximately one positive LE for each XIF neuron. This indicates in particular that a single XIF neuron renders the network dynamics chaotic. A simple mean-field approach, which can be justified by properties of the CLVs, explains the finding. As an application, we propose a spike-based computing scheme where our networks serve as computational reservoirs and their different stability properties yield different computational capabilities.

Lyapunov spectra and collective modes of chimera states in globally coupled Stuart-Landau oscillators

Oscillatory systems with long-range or global coupling offer promising insight into the interplay between high-dimensional (or microscopic) chaotic motion and collective interaction patterns. Within this paper, we use Lyapunov analysis to investigate whether chimera states in globally coupled Stuart-Landau (SL) oscillators exhibit collective degrees of freedom. We compare two types of chimera states, which emerge in SL ensembles with linear and nonlinear global coupling, respectively, the latter introducing a constraint that conserves the oscillation of the mean. Lyapunov spectra reveal that for both chimera states the Lyapunov exponents split into several groups with different convergence properties in the limit of large system size. Furthermore, in both cases the Lyapunov dimension is found to scale extensively and the localization properties of covariant Lypunov vectors manifest the presence of collective Lyapunov modes. Here, however, we find qualitative differences between the two types of chimera states: Whereas the ones in the system under nonlinear global coupling exhibit only slow collective modes corresponding to Lyapunov exponents equal or close to zero, those which experience the linear mean-field coupling exhibit also faster collective modes associated with Lyapunov exponents with large positive or negative values. Furthermore, for the fastest collective mode we showed that it spreads across both synchonous and incoherent oscillators.

Intermittent stickiness synchronization

This work uses the statistical properties of finite-time Lyapunov exponents (FTLEs) to investigate the intermittent stickiness synchronization (ISS) observed in the mixed phase space of high-dimensional Hamiltonian systems. Full stickiness synchronization (SS) occurs when all FTLEs from a chaotic trajectory tend to zero for arbitrarily long time windows. This behavior is a consequence of the sticky motion close to regular structures which live in the high-dimensional phase space and affects all unstable directions proportionally by the same amount, generating a kind of collective motion. Partial SS occurs when at least one FTLE approaches zero. Thus, distinct degrees of partial SS may occur, depending on the values of nonlinearity and coupling parameters, on the dimension of the phase space, and on the number of positive FTLEs. Through filtering procedures used to precisely characterize the sticky motion, we are able to compute the algebraic decay exponents of the ISS and to obtain remarkable evidence about the existence of a universal behavior related to the decay of time correlations encoded in such exponents. In addition we show that even though the probability of finding full SS is small compared to partial SSs, the full SS may appear for very long times due to the slow algebraic decay of time correlations in mixed phase space. In this sense, observations of very late intermittence between chaotic motion and full SS become rare events.

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.

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'.

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.

Self-Averaging Fluctuations in the Chaoticity of Simple Fluids

Bulk properties of equilibrium liquids are a manifestation of intermolecular forces. Here, we show how these forces imprint on dynamical fluctuations in the Lyapunov exponents for simple fluids with and without attractive forces. While the bulk of the spectrum is strongly self-averaging, the first Lyapunov exponent self-averages only weakly and at a rate that depends on the length scale of the intermolecular forces; short-range repulsive forces quantitatively dominate longer-range attractive forces, which act as a weak perturbation that slows the convergence to the thermodynamic limit. Regardless of intermolecular forces, the fluctuations in the Kolmogorov-Sinai entropy rate diverge, as one expects for an extensive quantity, and the spontaneous fluctuations of these dynamical observables obey fluctuation-dissipation-like relationships. Together, these results are a representation of the van der Waals picture of fluids and another lens through which we can view the liquid state.

Critical transitions and perturbation growth directions

Critical transitions occur in a variety of dynamical systems. Here we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions. As suitable indicator variables for critical transitions, we consider changes in growth rates and directions of covariant Lyapunov vectors. Studying critical transitions in several models of fast-slow systems, i.e., a network of coupled FitzHugh-Nagumo oscillators, models for Josephson junctions, and the Hindmarsh-Rose model, we find that tangencies between covariant Lyapunov vectors are a common and maybe generic feature during critical transitions. We further demonstrate that this deviation from hyperbolic dynamics is linked to the occurrence of critical transitions by using it as an indicator variable and evaluating the prediction success through receiver operating characteristic curves. In the presence of noise, we find the alignment of covariant Lyapunov vectors and changes in finite-time Lyapunov exponents to be more successful in announcing critical transitions than common indicator variables as, e.g., finite-time estimates of the variance. Additionally, we propose a new method for estimating approximations of covariant Lyapunov vectors without knowledge of the future trajectory of the system. We find that these approximated covariant Lyapunov vectors can also be applied to predict critical transitions.

From dynamical systems with time-varying delay to circle maps and Koopman operators

In this paper, we investigate the influence of the retarded access by a time-varying delay on the dynamics of delay systems. We show that there are two universality classes of delays, which lead to fundamental differences in dynamical quantities such as the Lyapunov spectrum. Therefore, we introduce an operator theoretic framework, where the solution operator of the delay system is decomposed into the Koopman operator describing the delay access and an operator similar to the solution operator known from systems with constant delay. The Koopman operator corresponds to an iterated map, called access map, which is defined by the iteration of the delayed argument of the delay equation. The dynamics of this one-dimensional iterated map determines the universality classes of the infinite-dimensional state dynamics governed by the delay differential equation. In this way, we connect the theory of time-delay systems with the theory of circle maps and the framework of the Koopman operator. In this paper, we extend our previous work [A. Otto, D. Müller, and G. Radons, Phys. Rev. Lett. 118, 044104 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.044104] by elaborating the mathematical details and presenting further results also on the Lyapunov vectors.

Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents

High-dimensional chaotic dynamical systems can exhibit strongly transient features. These are often associated with instabilities that have a finite-time duration. Because of the finite-time character of these transient events, their detection through infinite-time methods, e.g., long term averages, Lyapunov exponents or information about the statistical steady-state, is not possible. Here, we utilize a recently developed framework, the Optimally Time-Dependent (OTD) modes, to extract a time-dependent subspace that spans the modes associated with transient features associated with finite-time instabilities. As the main result, we prove that the OTD modes, under appropriate conditions, converge exponentially fast to the eigendirections of the Cauchy-Green tensor associated with the most intense finite-time instabilities. Based on this observation, we develop a reduced-order method for the computation of finite-time Lyapunov exponents (FTLE) and vectors. In high-dimensional systems, the computational cost of the reduced-order method is orders of magnitude lower than the full FTLE computation. We demonstrate the validity of the theoretical findings on two numerical examples.

Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics

The deterministic equations describing the dynamics of the atmosphere (and of the climate system) are known to display the property of sensitivity to initial conditions. In the ergodic theory of chaos, this property is usually quantified by computing the Lyapunov exponents. In this review, these quantifiers computed in a hierarchy of atmospheric models (coupled or not to an ocean) are analyzed, together with their local counterparts known as the local or finite-time Lyapunov exponents. It is shown in particular that the variability of the local Lyapunov exponents (corresponding to the dominant Lyapunov exponent) decreases when the model resolution increases. The dynamics of (finite-amplitude) initial condition errors in these models is also reviewed, and in general found to display a complicated growth far from the asymptotic estimates provided by the Lyapunov exponents. The implications of these results for operational (high resolution) atmospheric and climate modelling are also discussed.

Anomalous dynamics and the choice of Poincaré recurrence set

We investigate the dependence of Poincaré recurrence-time statistics on the choice of recurrence set by sampling the dynamics of two- and four-dimensional Hamiltonian maps. We derive a method that allows us to visualize the direct relation between the shape of a recurrence set and the values of its return probability distribution in arbitrary phase-space dimensions. Such a procedure, which is shown to be quite effective in the detection of tiny regions of regular motion, allows us to explain why similar recurrence sets have very different distributions and how to modify them in order to enhance their return probabilities. Applied to data, this enables us to understand the coexistence of extremely long, transient powerlike decays whose anomalous exponent depends on the chosen recurrence set.

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.

Numerical test for hyperbolicity of chaotic dynamics in time-delay systems

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them, previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.

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.

Estimating the Dimension of an Inertial Manifold from Unstable Periodic Orbits

We provide numerical evidence that a finite-dimensional inertial manifold on which the dynamics of a chaotic dissipative dynamical system lives can be constructed solely from the knowledge of a set of unstable periodic orbits. In particular, we determine the dimension of the inertial manifold for the Kuramoto-Sivashinsky system and find it to be equal to the "physical dimension" computed previously via the hyperbolicity properties of covariant Lyapunov vectors.

Characteristic distribution of finite-time Lyapunov exponents for chimera states

Our fascination with chimera states stems partially from the somewhat paradoxical, yet fundamental trait of identical, and identically coupled, oscillators to split into spatially separated, coherently and incoherently oscillating groups. While the list of systems for which various types of chimeras have already been detected continues to grow, there is a corresponding increase in the number of mathematical analyses aimed at elucidating the fundamental reasons for this surprising behaviour. Based on the model systems, there are strong indications that chimera states may generally be ubiquitous in naturally occurring systems containing large numbers of coupled oscillators - certain biological systems and high-Tc superconducting materials, for example. In this work we suggest a new way of detecting and characterising chimera states. Specifically, it is shown that the probability densities of finite-time Lyapunov exponents, corresponding to chimera states, have a definite characteristic shape. Such distributions could be used as signatures of chimera states, particularly in systems for which the phases of all the oscillators cannot be measured directly. For such cases, we suggest that chimera states could perhaps be detected by reconstructing the characteristic distribution via standard embedding techniques, thus making it possible to detect chimera states in systems where they could otherwise exist unnoticed.

Superlinearly scalable noise robustness of redundant coupled dynamical systems

We illustrate through theory and numerical simulations that redundant coupled dynamical systems can be extremely robust against local noise in comparison to uncoupled dynamical systems evolving in the same noisy environment. Previous studies have shown that the noise robustness of redundant coupled dynamical systems is linearly scalable and deviations due to noise can be minimized by increasing the number of coupled units. Here, we demonstrate that the noise robustness can actually be scaled superlinearly if some conditions are met and very high noise robustness can be realized with very few coupled units. We discuss these conditions and show that this superlinear scalability depends on the nonlinearity of the individual dynamical units. The phenomenon is demonstrated in discrete as well as continuous dynamical systems. This superlinear scalability not only provides us an opportunity to exploit the nonlinearity of physical systems without being bogged down by noise but may also help us in understanding the functional role of coupled redundancy found in many biological systems. Moreover, engineers can exploit superlinear noise suppression by starting a coupled system near (not necessarily at) the appropriate initial condition.

Manifold angles, the concept of self-similarity, and angle-enhanced bifurcation diagrams

Chaos and regularity are routinely discriminated by using Lyapunov exponents distilled from the norm of orthogonalized Lyapunov vectors, propagated during the temporal evolution of the dynamics. Such exponents are mean-field-like averages that, for each degree of freedom, squeeze the whole temporal evolution complexity into just a single number. However, Lyapunov vectors also contain a step-by-step record of what exactly happens with the angles between stable and unstable manifolds during the whole evolution, a big-data information permanently erased by repeated orthogonalizations. Here, we study changes of angles between invariant subspaces as observed during temporal evolution of Hénon’s system. Such angles are calculated numerically and analytically and used to characterize self-similarity of a chaotic attractor. In addition, we show how standard tools of dynamical systems may be angle-enhanced by dressing them with informations not difficult to extract. Such angle-enhanced tools reveal unexpected and practical facts that are described in detail. For instance, we present a video showing an angle-enhanced bifurcation diagram that exposes from several perspectives the complex geometrical features underlying the attractors. We believe such findings to be generic for extended classes of systems.

Cycles, randomness, and transport from chaotic dynamics to stochastic processes

An overview of advances at the frontier between dynamical systems theory and nonequilibrium statistical mechanics is given. Sensitivity to initial conditions is a mechanism at the origin of dynamical randomness-alias temporal disorder-in deterministic dynamical systems. In spatially extended systems, sustaining transport processes, such as diffusion, relationships can be established between the characteristic quantities of dynamical chaos and the transport coefficients, bringing new insight into the second law of thermodynamics. With methods from dynamical systems theory, the microscopic time-reversal symmetry can be shown to be broken at the statistical level of description in nonequilibrium systems. In this way, the thermodynamic entropy production turns out to be related to temporal disorder and its time asymmetry away from equilibrium.

Regeneration cycle and the covariant Lyapunov vectors in a minimal wall turbulence

Considering a wall turbulence as a chaotic dynamical system, we study regeneration cycles in a minimal wall turbulence from the viewpoint of orbital instability by employing the covariant Lyapunov analysis developed by [F. Ginelli et al. Phys. Rev. Lett. 99, 130601 (2007)]. We divide the regeneration cycle into two phases and characterize them with the local Lyapunov exponents and the covariant Lyapunov vectors of the Navier-Stokes turbulence. In particular, we show numerically that phase (i) is dominated by instabilities related to the sinuous mode and the streamwise vorticity, and there is no instability in phase (ii). Furthermore, we discuss a mechanism of the regeneration cycle, making use of an energy budget analysis.

Mechanism for stickiness suppression during extreme events in Hamiltonian systems

In this paper we study how hyperbolic and nonhyperbolic regions in the neighborhood of a resonant island perform an important role allowing or forbidding stickiness phenomenon around islands in conservative systems. The vicinity of the island is composed of nonhyperbolic areas that almost prevent the trajectory to visit the island edge. For some specific parameters tiny channels are embedded in the nonhyperbolic area that are associated to hyperbolic fixed points localized in the neighborhood of the islands. Such channels allow the trajectory to be injected in the inner portion of the vicinity. When the trajectory crosses the barrier imposed by the nonhyperbolic regions, it spends a long time abandoning the vicinity of the island, since the barrier also prevents the trajectory from escaping from the neighborhood of the island. In this scenario the nonhyperbolic structures are responsible for the stickiness phenomena and, more than that, the strength of the sticky effect. We show that those properties of the phase space allow us to manipulate the existence of extreme events (and the transport associated to it) responsible for the nonequilibrium fluctuation of the system. In fact we demonstrate that by monitoring very small portions of the phase space (namely, ≈1×10(-5)% of it) it is possible to generate a completely diffusive system eliminating long-time recurrences that result from the stickiness phenomenon.

Oseledets' splitting of standard-like maps

For the class of differentiable maps of the plane and, in particular, for standard-like maps (McMillan form), a simple relation is shown between the directions of the local invariant manifolds of a generic point and its contribution to the finite-time Lyapunov exponents (FTLE) of the associated orbit. By computing also the point-wise curvature of the manifolds, we produce a comparative study between local Lyapunov exponent, manifold's curvature and splitting angle between stable/unstable manifolds. Interestingly, the analysis of the Chirikov-Taylor standard map suggests that the positive contributions to the FTLE average mostly come from points of the orbit where the structure of the manifolds is locally hyperbolic: where the manifolds are flat and transversal, the one-step exponent is predominantly positive and large; this behaviour is intended in a purely statistical sense, since it exhibits large deviations. Such phenomenon can be understood by analytic arguments which, as a by-product, also suggest an explicit way to point-wise approximate the splitting.

Backward and covariant Lyapunov vectors and exponents for hard-disk systems with a steady heat current

The covariant Lyapunov analysis is generalized to systems attached to deterministic thermal reservoirs that create a heat current across the system and perturb it away from equilibrium. The change in the Lyapunov exponents as a function of heat current is described and explained. Both the nonequilibrium backward and covariant hydrodynamic Lyapunov modes are analyzed and compared. The movement of the converged angle between the hydrodynamic stable and unstable conjugate manifolds with the free flight time of the dynamics is accurately predicted for any nonequilibrium system simply as a function of their exponent. The nonequilibrium positive and negative LP mode frequencies are found to be asymmetrical, causing the negative mode to oscillate between the two functional forms of each mode in the positive conjugate mode pair. This in turn leads to the angular distributions between the conjugate modes to oscillate symmetrically about π/2 at a rate given by the difference between the positive and negative mode frequencies.

Predictable nonwandering localization of covariant Lyapunov vectors and cluster synchronization in scale-free networks of chaotic maps

Covariant Lyapunov vectors for scale-free networks of Hénon maps are highly localized. We revealed two mechanisms of the localization related to full and phase cluster synchronization of network nodes. In both cases the localization nodes remain unaltered in the course of the dynamics, i.e., the localization is nonwandering. Moreover, this is predictable: The localization nodes are found to have specific dynamical and topological properties and they can be found without computing of the covariant vectors. This is an example of explicit relations between the system topology, its phase-space dynamics, and the associated tangent-space dynamics of covariant Lyapunov vectors.

Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate

Motivated by recent experimental works, we investigate a system of vortex dynamics in an atomic Bose-Einstein condensate (BEC), consisting of three vortices, two of which have the same charge. These vortices are modeled as a system of point particles which possesses a Hamiltonian structure. This tripole system constitutes a prototypical model of vortices in BECs exhibiting chaos. By using the angular momentum integral of motion, we reduce the study of the system to the investigation of a two degree of freedom Hamiltonian model and acquire quantitative results about its chaotic behavior. Our investigation tool is the construction of scan maps by using the Smaller ALignment Index as a chaos indicator. Applying this approach to a large number of initial conditions, we manage to accurately and efficiently measure the extent of chaos in the model and its dependence on physically important parameters like the energy and the angular momentum of the system.

Manifold structures of unstable periodic orbits and the appearance of periodic windows in chaotic systems

Manifold structures of the Lorenz system, the Hénon map, and the Kuramoto-Sivashinsky system are investigated in terms of unstable periodic orbits embedded in the attractors. Especially, changes of manifold structures are focused on when some parameters are varied. The angle between a stable manifold and an unstable manifold (manifold angle) at every sample point along an unstable periodic orbit is measured using the covariant Lyapunov vectors. It is found that the angle characterizes the parameter at which the periodic window corresponding to the unstable periodic orbit finishes, that is, a saddle-node bifurcation point. In particular, when the minimum value of the manifold angle along an unstable periodic orbit at a parameter is small (large), the corresponding periodic window exists near (away from) the parameter. It is concluded that the window sequence in a parameter space can be predicted from the manifold angles of unstable periodic orbits at some parameter. The fact is important because the local information in a parameter space characterizes the global information in it. This approach helps us find periodic windows including very small ones.

Nonlinear stability of traffic models and the use of Lyapunov vectors for estimating the traffic state

Valuable information for estimating the traffic flow is obtained with current GPS technology by monitoring position and velocity of vehicles. In this paper, we present a proof of concept study that shows how the traffic state can be estimated using only partial and noisy data by assimilating them in a dynamical model. Our approach is based on a data assimilation algorithm, developed by the authors for chaotic geophysical models, designed to be equivalent but computationally much less demanding than the traditional extended Kalman filter. Here we show that the algorithm is even more efficient if the system is not chaotic and demonstrate by numerical experiments that an accurate reconstruction of the complete traffic state can be obtained at a very low computational cost by monitoring only a small percentage of vehicles.

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.

Covariant Lyapunov vectors from reconstructed dynamics: the geometry behind true and spurious Lyapunov exponents

The estimation of Lyapunov exponents from time series suffers from the appearance of spurious Lyapunov exponents due to the necessary embedding procedure. Separating true from spurious exponents poses a fundamental problem which is not yet solved satisfactorily. We show, in this Letter, analytically and numerically that covariant Lyapunov vectors associated with true exponents lie in the tangent space of the reconstructed attractor. Therefore, we use the angle between the covariant Lyapunov vectors and the tangent space of the reconstructed attractor to identify the true Lyapunov exponents. The usefulness of our method, also for noisy situations, is demonstrated by applications to data from model systems and a NMR laser experiment.

Anomalous transport induced by nonhyperbolicity

In this paper we study how deterministic features presented by a system can be used to perform direct transport in a quasisymmetric potential and weak dissipative system. We show that the presence of nonhyperbolic regions around acceleration areas of the phase space plays an important role in the acceleration of particles giving rise to direct transport in the system. Such an effect can be observed for a large interval of the weak asymmetric potential parameter allowing the possibility to obtain useful work from unbiased nonequilibrium fluctuation in real systems even in a presence of a quasisymmetric potential.