Ergodic theory of chaos and strange attractors

DOI10.1103/RevModPhys.57.617zbMath0989.37516OpenAlexW1998727969WikidataQ21698194 ScholiaQ21698194MaRDI QIDQ4537864

David Pierre Ruelle, Jean-Pierre Eckmann

Publication date: 9 July 2002

Published in: Reviews of Modern Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1103/revmodphys.57.617

Mathematics Subject Classification ID

Dynamical systems in fluid mechanics, oceanography and meteorology (37N10) Entropy and other invariants, isomorphism, classification in ergodic theory (37A35) Strange attractors, chaotic dynamics of systems with hyperbolic behavior (37D45) Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory (37-02)

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Examples from field theory ⋮ A discrete time neural network model with spiking neurons ⋮ A novel hyperchaos system only with one equilibrium ⋮ Stochastic suspensions of heavy particles ⋮ Climate dynamics and fluid mechanics: Natural variability and related uncertainties ⋮ Sudden shifts in ecological systems: intermittency and transients in the coupled Ricker population model ⋮ New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems ⋮ Inductorless Chua's circuit: experimental time series analysis ⋮ Multi-scale entropy analysis of Mississippi river flow ⋮ Short-time linear response with reduced-rank tangent map ⋮ Parametric characterisation of a chaotic attractor using the two scale Cantor measure ⋮ Chaotic time series analysis ⋮ Emergent universe scenario in the Einstein-Gauss-Bonnet gravity with dilaton ⋮ Nonequilibrium thermodynamics of the first and second kind: averages and fluctuations ⋮ Physical measures for multivalued inverse iterates near hyperbolic repellors ⋮ Dimension theory for invariant measures of endomorphisms ⋮ Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator ⋮ On the concept of stationary Lyapunov basis ⋮ A chaotic Hopf bifurcation in coupled maps ⋮ Infinite-dimensional SRB measures ⋮ Dynamics of coarse-grained variables in spatially extended systems ⋮ The topological skeleton of cellular automaton dynamics ⋮ A hybrid of fractal image coding and fractal dimension for an efficient retrieval method ⋮ Semiquantum chaos and the large \(N\) expansion ⋮ Using neural nets to look for chaos ⋮ Testing for nonlinearity in time series: the method of surrogate data ⋮ Problems in estimating dynamics from data ⋮ Dynamical signatures ⋮ Hierarchical resolution of power spectra ⋮ Invariant measure in coupled maps ⋮ Effect of bounded noise on chaotic motions of stochastically perturbed slowly varying oscillator ⋮ De-synchronization and chaos in two inductively coupled Van der Pol auto-generators ⋮ Localized Lyapunov exponents and the prediction of predictability ⋮ Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbits ⋮ Analysing spatially extended high-dimensional dynamics by recurrence plots ⋮ Information-geometric indicators of chaos in Gaussian Models on statistical manifolds of negative Ricci Curvature ⋮ Effects of manifold correction methods on chaos indicators ⋮ Statistical mechanics of complex systems for pattern identification ⋮ A theorem on the first heteroclinic tangency in two-dimensional maps. Orientation-preserving cases ⋮ Phase-transition-like phenomenon in a piecewise linear map ⋮ The dimension spectrum of Axiom A attractors ⋮ Hausdorff dimensions in two-dimensional maps and thermodynamic formalism ⋮ Convergence of the \(K_ 2\) entropy for random noises with power law spectra ⋮ Horseshoe chaos and topological entropy estimate in a simple power system ⋮ Sources of complex dynamics in two-sector growth models ⋮ Some aspects of the Liouville equation in mathematical physics and statistical mechanics ⋮ Small noise asymptotics for invariant densities for a class of diffusions: a control theoretic view ⋮ Scaling properties of a hybrid Fermi-Ulam-bouncer model ⋮ Phase transition in dynamical systems: defining classes of universality for two-dimensional Hamiltonian mappings via critical exponents ⋮ Can drag force suppress Fermi acceleration in a bouncer model? ⋮ Reliability of coupled oscillators ⋮ Variational approach to closure of nonlinear dynamical systems: autonomous case ⋮ On temporal scale separation in coupled data assimilation with the ensemble Kalman filter ⋮ Statistical significance of small ensembles of simulations and detection of the internal climate variability: an excitable ocean system case study ⋮ The theory of parallel climate realizations. A new framework of ensemble methods in a changing climate: an overview ⋮ Response and sensitivity using Markov chains ⋮ On stochastic dynamic analysis and assessment of bistable structures ⋮ Dynamical transitions of the quasi-periodic plasma model ⋮ The stability of fixed points on switching manifolds of piecewise-smooth continuous maps ⋮ The metric entropy of endomorphisms ⋮ Long-range order in the scaling behaviour of hyperbolic dynamical systems ⋮ Fractal dimension of spatially extended systems ⋮ A maximum likelihood approach to correlation dimension and entropy estimation ⋮ Calculation of Lyapunov exponents for dynamic systems with discontinuities. ⋮ Chaos and randomness: An equivalence proof of a generalized version of the Shannon entropy and the Kolmogorov-Sinai entropy for Hamiltonian dynamical systems ⋮ ``Linear chaos via paradoxical set decompositions. ⋮ A mechanism for the onset of chaos in mechanical systems with motion-limiting stops. ⋮ Wavelet shrinkage of a noisy dynamical system with non-linear noise impact ⋮ Control of chaotic systems using an on-line trained linear neural controller ⋮ Competition, collusion, and chaos ⋮ Estimating the correlation dimension of an attractor from noisy and small datasets based on re-embedding ⋮ A practical method for calculating largest Lyapunov exponents from small data sets ⋮ High level chaos in the exchange and index markets ⋮ A comparative study of pattern synchronization detection between neural signals using different cross-entropy measures ⋮ Individual-based chaos: extensions of the discrete logistic model ⋮ Chaos of a new class of Hopfield neural networks ⋮ Dynamical quantities and their numerical analysis by saddle periodic orbits ⋮ Jacobi fields on statistical manifolds of negative curvature ⋮ Periodic orbits and long transients in coupled map lattices ⋮ Chaotic attractors of a locally conservative hyperbolic map with overlap ⋮ Rate of escape of some chaotic Julia sets ⋮ Superstable cycles and magnetization plateaus for antiferromagnetic spin-1 Ising and Ising-Heisenberg models on diamond chains ⋮ Parametric smoothness and self-scaling of the statistical properties of a minimal climate model: what beyond the mean field theories? ⋮ The nonlinear dynamics of the damped and driven Toda chain. II: Fourier and Lyapunov analysis of tori ⋮ Effect of seasonality on the dynamics of 2 and 3 species prey\,-\,predator systems ⋮ Is the largest Lyapunov exponent preserved in embedded dynamics? ⋮ Robust chaos in neural networks ⋮ Encoding \(n\)-to-1 Baker's transformations ⋮ Hausdorff dimension of chaotic attractors in a class of nonsmooth systems ⋮ Low-frequency variability and heat transport in a low-order nonlinear coupled ocean-atmosphere model ⋮ Three-dimensional forced-damped dynamical systems with rich dynamics: bifurcations, chaos and unbounded solutions ⋮ Numerical simulations of the humid atmosphere above a mountain ⋮ Rich dynamics in some discrete-time car-following models ⋮ Synchronism in population networks with non-linear coupling ⋮ Ledrappier-Young formula and exact dimensionality of self-affine measures ⋮ Absolute continuity of stable foliations for mappings of Banach spaces ⋮ Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers ⋮ Strange attractors in periodically kicked predator-prey system with discrete and distributed delay ⋮ Subadditive and multiplicative ergodic theorems ⋮ Leading order response of statistical averages of a dynamical system to small stochastic perturbations ⋮ Coexistence of strange nonchaotic attractors and a special mixed attractor caused by a new intermittency in a periodically driven vibro-impact system ⋮ Bifurcation in the almost periodic \(2\)D Ricker map ⋮ The evolution of developmental timing in natural enemy systems ⋮ Estimating Lyapunov exponents on a noisy environment by global and local Jacobian indirect algorithms ⋮ Chaotic dynamics of string around charged black brane with hyperscaling violation ⋮ Conservative generalized bifurcation diagrams and phase space properties for oval-like billiards ⋮ Chaos analysis and explicit series solutions to the seasonally forced SIR epidemic model ⋮ Cumulant analysis of the statistical properties of a deterministically thermostated harmonic oscillator ⋮ Particle filters with nudging in multiscale chaotic systems: with application to the Lorenz '96 atmospheric model ⋮ How to perturb Bernoulli shift map ⋮ A second order phase transition characterized in the suppression of unlimited chaotic diffusion for a dissipative standard mapping ⋮ Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems ⋮ Lyapunov mode dynamics in hard-disk systems ⋮ Spatial structure of stationary nonequilibrium states in the thermostatted periodic Lorentz gas ⋮ Detecting regular dynamics from time series using permutations slopes ⋮ Invariant measures in non-conformal fibered systems with singularities ⋮ Chaos in integer order and fractional order financial systems and their synchronization ⋮ The impact of tariff policies on vaccine supply chains: short-term and evolutionary game behaviors based on uncertain utility ⋮ Multifractality in intracellular enzymatic reactions ⋮ A selection procedure for extracting the unique Feller weak solution of degenerate diffusions ⋮ Search for optimal parameters in a recurrence analysis of the Duffing system with varying damping ⋮ Contribution of individual degrees of freedom to Lyapunov vectors in many-body systems ⋮ Nonparametric neural network estimation of Lyapunov exponents and a direct test for chaos ⋮ In search of chaos and complexity of a cognitive language-learning system ⋮ Adjoint sensitivity analysis of chaotic systems using cumulant truncation ⋮ Permutation entropy revisited ⋮ Numerical convergence of the Lyapunov spectrum computed using low Mach number solvers ⋮ Chimeras in multivariable coupled Rössler oscillators ⋮ Causation entropy identifies indirect influences, dominance of neighbors and anticipatory couplings ⋮ Chaotic attractors in the four-dimensional Leslie-Gower competition model ⋮ Population distribution and synchronized dynamics in a metapopulation model in two geographic scales ⋮ Rich dynamics in some generalized difference equations ⋮ Counting unstable periodic orbits in noisy chaotic systems: A scaling relation connecting experiment with theory ⋮ Dynamical ensembles in nonequilibrium statistical mechanics and their representations ⋮ Chaos in a dynamic model of traffic flows in an origin-destination network ⋮ Easy-to-implement method to target nonlinear systems ⋮ Practical implementation of nonlinear time series methods: The <scp>TISEAN</scp> package ⋮ Estimating the amplitude of measurement noise present in chaotic time series ⋮ Statistical physics approaches to the complex Earth system ⋮ When good theories make bad predictions ⋮ Probability of second law violations in shearing steady states ⋮ Analyzing multiple nonlinear time series with extended Granger causality ⋮ Study on statistical properties of chaotic laser light ⋮ A novel moment approach for calculation of the Perron-Frobenius spectrum ⋮ Hyperchaotic behavior in a non-autonomous unified chaotic system with continuous periodic switch ⋮ Property of period-doubling bifurcation cascades of discrete dynamical systems ⋮ Local prediction of nonlinear time series using support vector regression ⋮ Solvable chaos ⋮ Chaos synchronization in long-range coupled map lattices ⋮ Statistical complexity and generalized number system ⋮ Lyapunov modes in hard-disk systems ⋮ Computing Lyapunov exponents based on the solution expression of the variational system ⋮ Computation of the Lyapunov spectrum for continuous-time dynamical systems and discrete maps ⋮ Natural tangent dynamics with recurrent biorthonormalizations: a geometric computational approach to dynamical systems exhibiting slow manifolds and periodic/chaotic limit sets ⋮ Invariant trajectory classification of dynamical systems with a case study on ECG ⋮ Optimization approach to the estimation and control of Lyapunov exponents ⋮ Measuring statistical dependences in a time series ⋮ Using filtering effects to identify objects ⋮ Numerical analysis on the complex dynamics in a chemical system ⋮ Metastability in stochastic replicator dynamics ⋮ von Neumann mutual information for anisotropic coupled oscillators interacting with a single two-level atom ⋮ On observer-based secure communication design using discrete-time hyperchaotic systems ⋮ Synchronization of fractional-order discrete-time chaotic systems by an exact delayed state reconstructor: application to secure communication ⋮ Detection of chaotic determinism in time series from randomly forced maps ⋮ Theoretical time series analysis from electric field oscillations generated by rate equations of generation-recombination processes in \(n\)-type semiconductors ⋮ Digital memcomputing: from logic to dynamics to topology ⋮ Testing for chaos in deterministic systems with noise ⋮ Coarse-Graining Autonomous ODE Systems by Inducing a Separation of Scales: Practical Strategies and Mathematical Questions ⋮ Multifractal structure of the harmonic measure of diffusion-limited aggregates ⋮ Non-strange chaotic attractors equivalent to their templates ⋮ LONG-RUN OPTIMAL BEHAVIOR IN A TWO-SECTOR ROBINSON–SOLOW–SRINIVASAN MODEL ⋮ On the Amplitude of External Perturbation and Chaos via Devil's Staircase — Stability of Attractors ⋮ APERIODIC DYNAMICS IN THE BERGSTROM/WYMER MODEL OF THE UNITED KINGDOM ⋮ Dimension theory of iterated function systems ⋮ Scattering theory and thermodynamics of quantum transport ⋮ A Modified 0-1 Test for Chaos Detection in Oversampled Time Series Observations ⋮ Chaos and noise ⋮ Scalar Decay in Chaotic Mixing ⋮ Estimating the largest Lyapunov exponent and noise level from chaotic time series ⋮ Two-particle circular billiards versus randomly perturbed one-particle circular billiards ⋮ SCALE-INVARIANCE OF NORMALIZED YEARLY MEAN GRAIN YIELD ANOMALY SERIES ⋮ CHAOS AND HYPERCHAOS IN A CLASS OF SIMPLE CELLULAR NEURAL NETWORKS MODELED BY O.D.E. ⋮ ATTRACTOR MODELING AND EMPIRICAL NONLINEAR MODEL REDUCTION OF DISSIPATIVE DYNAMICAL SYSTEMS ⋮ CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS ⋮ CHAOTIC DYNAMICS OF A FIVE-DIMENSIONAL NONLINEAR NETWORK ⋮ NONLINEAR TIME SERIES ANALYSIS IN EPILEPSY ⋮ Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model ⋮ Numerical Analysis and Improved Algorithms for Lyapunov-Exponent Calculation of Discrete-Time Chaotic Systems ⋮ A Very Simple Method to Calculate the (Positive) Largest Lyapunov Exponent Using Interval Extensions ⋮ Approximate Entropy as an Irregularity Measure for Financial Data ⋮ WHAT CONNECTION TOPOLOGY PROHIBIT CHAOS IN CONTINUOUS TIME NETWORKS? ⋮ Dynamics of Oscillator Chains ⋮ Role of Chaos for the Validity of Statistical Mechanics Laws: Diffusion and Conduction ⋮ Synchronization in networks with random interactions: Theory and applications ⋮ Dimensionality and morphology of particle and bubble clusters in turbulent flow ⋮ Complex dynamics in simple Hopfield neural networks ⋮ Strange nonchaotic attractors in Harper maps ⋮ Towards better DNA chip hybridization using chaotic advection ⋮ SCALING BEHAVIOR OF DIFFUSION LIMITED AGGREGATION IN PERCOLATION CLUSTER ⋮ HORSESHOE IN THE HYPERCHAOTIC MCK CIRCUIT ⋮ Routes to chaos in wide-gap spherical Couette flow ⋮ A Lagrangian analysis of advection-diffusion equation for a three dimensional chaotic flow ⋮ Experimental study of Taylor’s hypothesis in a turbulent soap film ⋮ Empiric Stochastic Stability of Physical and Pseudo-physical Measures ⋮ Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics ⋮ Singularities of the susceptibility of a Sinai–Ruelle–Bowen measure in the presence of stable–unstable tangencies ⋮ The ensemble of random Markov matrices ⋮ Independent coordinates for strange attractors from mutual information ⋮ KAGOME APPROXIMATION FOR ³He ON HUSIMI LATTICE WITH TWO- AND THREE-SITE EXCHANGE INTERACTIONS ⋮ Switching contact task control in hydraulic actuators: Stability analysis and experimental evaluation ⋮ Path probabilities of continuous time random walks ⋮ Fluctuating observation time ensembles in the thermodynamics of trajectories ⋮ Rare-event trajectory ensemble approach to study dynamical phase transitions in the zero temperature Glauber model ⋮ Absence of dissipation in trajectory ensembles biased by currents ⋮ Effective dynamics in an asymmetric death-branching process ⋮ Chaotic attractors in Atkinson–Allen model of four competing species ⋮ Study of a chaotic mixing system for DNA chip hybridization chambers ⋮ Clustering and collisions of heavy particles in random smooth flows ⋮ PSEUDO RANDOM NUMBER GENERATOR BASED ON SYNCHRONIZED CHAOTIC MAPS ⋮ CHAOS, BIFURCATION AND ROBUSTNESS OF A CLASS OF HOPFIELD NEURAL NETWORKS ⋮ A SIMPLE METHOD FOR FINDING TOPOLOGICAL HORSESHOES ⋮ HYPERCHAOS IN A SIMPLE CNN ⋮ Connecting ergodicity and dimension in dynamical systems ⋮ DYNAMICAL CHAOS AND NONEQUILIBRIUM STATISTICAL MECHANICS ⋮ Analytical study of the superstable 3-cycle in the logistic map ⋮ HYPERCHAOTIC SYNCHRONIZATION ⋮ The curvature index and synchronization of dynamical systems ⋮ Extreme value theory for singular measures ⋮ Characterization of multiscroll attractors using Lyapunov exponents and Lagrangian coherent structures ⋮ Relation between delayed feedback and delay-coupled systems and its application to chaotic lasers ⋮ Asymptotic transversal stability for synchronized attractors in a metapopulation model ⋮ CHAOTIC CONTROL OF LIU SYSTEM WITH PERIODIC PARAMETRIC PERTURBATIONS ⋮ Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise ⋮ Topological analysis of chaotic dynamical systems ⋮ Chaotic Polynomial Maps ⋮ Deciphering chaos in evolutionary games ⋮ Quantitative measure of memory loss in complex spatiotemporal systems ⋮ Using machine learning to predict statistical properties of non-stationary dynamical processes: System climate,regime transitions, and the effect of stochasticity ⋮ Two chaotic global models for cereal crops cycles observed from satellite in northern Morocco ⋮ Complex statistics and diffusion in nonlinear disordered particle chains ⋮ Attractor comparisons based on density ⋮ Log-corrected energy spectrum and dimension of attractor in two-dimensional turbulence ⋮ On a class of stable conditional measures ⋮ New variational principles for locating periodic orbits of differential equations ⋮ Unstable periodic orbits in the Lorenz attractor ⋮ CONSTRUCTING CHAOTIC POLYNOMIAL MAPS ⋮ A FAST ALGORITHM FOR COMPUTING SAMPLE ENTROPY ⋮ HIGH-DIMENSIONAL CHAOS IN DISSIPATIVE AND DRIVEN DYNAMICAL SYSTEMS ⋮ Extreme value laws for dynamical systems under observational noise ⋮ Unnamed Item ⋮ Application of Indicators for Chaos in Chaotic Circuit Systems ⋮ A NEURO-FUZZY BASED MODEL FOR ACCURATE ESTIMATION OF THE LYAPUNOV EXPONENTS OF AN UNKNOWN DYNAMICAL SYSTEM ⋮ CHAOS AND WEATHER FORECASTING: THE ROLE OF THE UNSTABLE SUBSPACE IN PREDICTABILITY AND STATE ESTIMATION PROBLEMS ⋮ Master-equation approach to deterministic chaos ⋮ Pointwise dimension, entropy and Lyapunov exponents for $C^{1}$ maps ⋮ AN APPLICATION OF ADOMIAN DECOMPOSITION FOR ANALYSIS OF FRACTIONAL-ORDER CHAOTIC SYSTEMS ⋮ ASYMPTOTIC TIME DECAY IN QUANTUM PHYSICS: A SELECTIVE REVIEW AND SOME NEW RESULTS ⋮ LYAPUNOV-EXPONENT SPECTRUM FROM NOISY TIME SERIES ⋮ A survey on transitivity in discrete time dynamical systems. application to symbolic systems and related languages ⋮ TOPOLOGICAL COMPLEXITY AND PREDICTABILITY IN THE DYNAMICS OF A TUMOR GROWTH MODEL WITH SHILNIKOV'S CHAOS ⋮ MODEL IMPERFECTION AND PREDICTING PREDICTABILITY ⋮ SUPREME LOCAL LYAPUNOV EXPONENTS AND CHAOTIC IMPULSIVE SYNCHRONIZATION ⋮ On the C²-creation of links of critical points ⋮ Sinai–Ruelle–Bowen measures for N-dimensional derived from Anosov diffeomorphisms ⋮ Discrete dynamical systems embedded in Cantor sets ⋮ Dynamics, stability, and statistics on lattices and networks ⋮ Random paths and current fluctuations in nonequilibrium statistical mechanics ⋮ A neural network model as a globally coupled map and applications based on chaos ⋮ Topology of trajectories of the 2D Navier–Stokes equations ⋮ Statistical properties of chaos demonstrated in a class of one-dimensional maps ⋮ Traveling waves in lattice models of multidimensional and multicomponent media. II. Ergodic properties and dimension ⋮ Control of dynamical systems behavior by parametric perturbations: An analytic approach ⋮ The nonequilibrium Lorentz gas ⋮ Enlarged scaling ranges for the KS‐entropy and the information dimension ⋮ Phase-space prediction of chaotic time series ⋮ Instability of pole solutions for planar propagating flames in sufficiently large domains ⋮ Measurements of a Physical Process Satisfy a Difference Equation ⋮ On quantum characteristic exponents ⋮ Liapunov exponents of simple smooth flows ⋮ Short-time Lyapunov exponent analysis and the transition to chaos in Taylor–Couette flow ⋮ Ergodicity and invariant measures of some randomly perturbed classical fields ⋮ Testing dynamical system variables for reconstruction ⋮ Fluctuation-response relations in systems with chaotic behavior ⋮ Chaotic and non-chaotic strange attractors of a class of non-autonomous systems ⋮ Entropy-based generating Markov partitions for complex systems ⋮ Matlab Code for Lyapunov Exponents of Fractional-Order Systems ⋮ Crisis of the chaotic attractor of a climate model: a transfer operator approach ⋮ Types of convergence for sequences of probability densities and attractors ⋮ Sample and moment lyapunov exponents of discrete linear dynamical systems ⋮ The dimension of attractors underlying periodic turbulent Poiseuille flow ⋮ A numerical method of K-S entropy calculation for a strange attractor ⋮ Random attractors for the three dimensional stochastical planetary geostrophic equations of large-scale ocean circulation ⋮ Delayed feedback control and phase reduction of unstable quasi-periodic orbits ⋮ Bifurcation behaviors of synchronized regions in logistic map networks with coupling delay ⋮ Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism ⋮ Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodic sets ⋮ Series-nonuniform rational B-spline signal feedback: From chaos to any embedded periodic orbit or target point ⋮ Cycles, randomness, and transport from chaotic dynamics to stochastic processes ⋮ Nonlinear time-series analysis revisited ⋮ Synchronization of chaotic systems ⋮ Computation of entropy and Lyapunov exponent by a shift transform ⋮ Dynamics of the stochastic Lorenz chaotic system with long memory effects ⋮ Stability and approximation of invariant measures for a class of nonexpanding transformations ⋮ Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator ⋮ Torus Breakdown and Homoclinic Chaos in a Glow Discharge Tube ⋮ Digit replacement: A generic map for nonlinear dynamical systems ⋮ Nuisance parameter free properties of correlation integral based statistics ⋮ Equivalence of physical and SRB measures in random dynamical systems ⋮ Kac–Rice fixed point analysis for single- and multi-layered complex systems ⋮ A novel method based on the pseudo-orbits to calculate the largest Lyapunov exponent from chaotic equations ⋮ On the construction of Hamiltonians ⋮ Towards a classification of generic bifurcations in dissipative dynamical systems ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Random attractor for the stochastic Cahn–Hilliard–Navier–Stokes system with small additive noise ⋮ Calculation of Hamilton energy and control of dynamical systems with different types of attractors ⋮ Chaotic motion of an array of vortices above a flat wall ⋮ NEW RESULTS ON THE ENTROPY OF DETERMINISTIC MAPS: APPLICATIONS TO LIAPUNOV EXPONENT, DIVERGENCE, COMPOSITIONAL INFERENCE, AND PATTERN ENCODING ⋮ Dimension, recurrence via entropy and Lyapunov exponents for map with singularities ⋮ On the limits of probabilistic forecasting in nonlinear time series analysis II: Differential entropy ⋮ Data-adaptive harmonic spectra and multilayer Stuart-Landau models ⋮ Measuring Lyapunov exponents of large chaotic systems with global coupling by time series analysis ⋮ Identifying the linear region based on machine learning to calculate the largest Lyapunov exponent from chaotic time series ⋮ Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos ⋮ Takens embedding theorems for forced and stochastic systems ⋮ Applications of bivariate and univariate local lyapunov exponents ⋮ Informational Entropy of Distributed Deterministic Maps: Applications to Small Samples of Fuzzy Sets and to Pattern Recognition ⋮ Recent Developments in Chaotic Time Series Analysis ⋮ RECONSTRUCTING DIFFERENTIAL EQUATION FROM A TIME SERIES ⋮ DYNAMICAL ANALYSIS OF A CHAOTIC SYSTEM WITH TWO DOUBLE-SCROLL CHAOTIC ATTRACTORS ⋮ A NONLINEAR DIFFERENTIAL EQUATION OF FRACTIONAL ORDER WITH CHAOTIC PROPERTIES ⋮ CLASSIFICATION OF CHAOTIC SEQUENCES WITH OPEN-LOOP ESTIMATOR — OPTIMAL DESIGN FOR NOISY ENVIRONMENTS ⋮ Revealing oscillators calculation of the dimension of attractors ⋮ Fundamental concepts of classical chaos I ⋮ IMPLEMENTING MEMRISTOR BASED CHAOTIC CIRCUITS ⋮ CRISIS-INDUCED INTERMITTENCY IN HAMILTONIAN SYSTEMS ⋮ A VIEW OF NEURAL NETWORKS AS DYNAMICAL SYSTEMS ⋮ ON THE STATISTICAL DISTRIBUTION OF FIRST-RETURN TIMES OF BALLS AND CYLINDERS IN CHAOTIC SYSTEMS ⋮ Sample Entropy Statistics and Testing for Order in Complex Physiological Signals ⋮ The Road to Chaos by Time-Asymmetric Hebbian Learning in Recurrent Neural Networks ⋮ A numerical Eulerian approach to mixing by chaotic advection ⋮ Lagrangian chaos, Eulerian chaos, and mixing enhancement in converging–diverging channel flows ⋮ ALGEBRAIC DYNAMICS AND ALGEBRAIC ENTROPY ⋮ Dynamical properties of a dissipative hybrid Fermi-Ulam-bouncer model ⋮ Interactions between short-term and long-term cardiovascular control mechanisms ⋮ Periodic orbit analysis at the onset of the unstable dimension variability and at the blowout bifurcation ⋮ Optimal instruments and models for noisy chaos ⋮ PARTITION-BASED ENTROPIES OF DETERMINISTIC AND STOCHASTIC MAPS ⋮ TIME SERIES ANALYSIS IN RECONSTRUCTED STATE SPACES ⋮ UNIFORM LYAPUNOV EXPONENTS ARE REALISED BY ERGODIC INVARIANT MEASURES ⋮ Dimensional characteristics of invariant measures for circle diffeomorphisms ⋮ Unnamed Item ⋮ Dynamical Analysis of Time Series by Statistical Tests ⋮ Structure and Breakdown of Invariant Tori in a 4-D Mapping Model of Accelerator Dynamics ⋮ Experimental Nonlinear Physics ⋮ Lyapunov exponents from observed time series ⋮ Determination of correlation spectra in chaotic systems ⋮ Pseudoperiodic driving: Eliminating multiple domains of attraction using chaos ⋮ Chaotic quantum phenomena without classical counterpart ⋮ Reduced Tangent Dynamics and Lyapunov Spectrum for Hamiltonian Systems ⋮ STATISTICAL ESTIMATION OF THE EMBEDDING DIMENSION OF A DYNAMICAL SYSTEM ⋮ Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy methods for sums of Lyapunov exponents for dilute gases ⋮ Probabilistic and thermodynamic aspects of dynamical systems ⋮ An analytical construction of the SRB measures for Baker-type maps ⋮ The conjugate-pairing rule for non-Hamiltonian systems ⋮ Scaling and interleaving of subsystem Lyapunov exponents for spatio-temporal systems ⋮ Entropy computing via integration over fractal measures ⋮ A novel image encryption algorithm based on chaotic billiards ⋮ Singular continuous spectra and discrete wave packet dynamics ⋮ HYPERCHAOS IN A NEW FAMILY OF SIMPLE CNNS ⋮ Numerical stability and error analysis for the incompressible Navier-Stokes equations ⋮ Interscale entanglement production in a quantum system simulating classical chaos ⋮ Redefinition of the Boltzmann–Gibbs–Shannon entropy in systems with continuous states ⋮ Unmixing in random flows ⋮ Rocard’s 1941 Chaotic Relaxation Econometric Oscillator ⋮ Modern Koopman Theory for Dynamical Systems ⋮ The information topology and true laminations for diffeomorphisms ⋮ Ultra-Chaos: An Insurmountable Objective Obstacle of Reproducibility and Replication ⋮ Liapunov stability and adding machines ⋮ Padé-like approximation and maximum entropy formalism applied to chaotic systems ⋮ Random attractors for the two-dimensional stochastic g-Navier-Stokes equations ⋮ Hidden spatiotemporal symmetries and intermittency in turbulence ⋮ Thermodynamic formalism for invariant measures in iterated function systems with overlaps ⋮ Invariant Sets in Quasiperiodically Forced Dynamical Systems ⋮ Period-Bubbling Transition to Chaos in Thermo-Viscoelastic Fluid Systems ⋮ Permutation Entropy of State Transition Networks to Detect Synchronization ⋮ New methods and structures in the theory of the multi-mode Dicke laser model ⋮ Smoothing data with local instabilities for the identification of chaotic systems ⋮ On the stochastic representation and Markov approximation of Hamiltonian systems ⋮ A probabilistic Takens theorem ⋮ The structure of reconstructed flows in latent spaces ⋮ The extended Granger causality analysis for Hodgkin–Huxley neuronal models ⋮ Stochastic approach to entropy production in chemical chaos ⋮ Estimating invariant measures and Lyapunov exponents ⋮ Scattering approach to the thermodynamics of quantum transport ⋮ Dynamic interlinking between near- and far-field wakes behind a pitching–heaving airfoil ⋮ Chaos, entanglement, and decoherence in the quantum kicked top ⋮ Characterization of nonlinear dynamic systems for engineering purposes – a partial review ⋮ HYPERBOLIC-LIKE PROPERTIES OF POPP'S ATTRACTOR ⋮ CRITICAL EXPONENTS AND SCALING PROPERTIES FOR THE CHAOTIC DYNAMICS OF A PARTICLE IN A TIME-DEPENDENT POTENTIAL BARRIER ⋮ MATRIX QUANTIZATION OF TURBULENCE ⋮ ESTIMATING PROBABILITY DENSITY FUNCTIONS AND ENTROPIES OF CHUA'S CIRCUIT USING B-SPLINE FUNCTIONS ⋮ Matlab Code for Lyapunov Exponents of Fractional-Order Systems, Part II: The Noncommensurate Case ⋮ A concept of homeomorphic defect for defining mostly conjugate dynamical systems ⋮ ENTROPY AND COMPLEXITY ANALYSIS OF INTRACRANIALLY RECORDED EEG ⋮ On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity ⋮ Generalized entropies of chaotic maps and flows: A unified approach ⋮ Entropy potential and Lyapunov exponents ⋮ Dynamics of coupled maps with a conservation law ⋮ On Overparametrization of Nonlinear Discrete Systems ⋮ Stable State Analysis of an Immune Network Model ⋮ Characteristic Scales of Reconstruction Distortions ⋮ Noise Induced Order: Stochastic Resonance ⋮ Entropy Analysis of Noise Contaminated Sequences ⋮ Adaptive strategies for recognition, noise filtering, control, synchronization and targeting of chaos ⋮ Chaotic dynamics of sea clutter ⋮ ABSENCE OF CHAOS AND 1/f SPECTRA, BUT EVIDENCE OF TAR NONLINEARITIES, IN THE CANADIAN EXCHANGE RATE ⋮ The Rényi entropy function and the large deviation of short return times ⋮ Chaotic Motions of a Duffing Oscillator Subjected to Combined Parametric and Quasiperiodic Excitation ⋮ Identification and reconstruction of chaotic systems using multiresolution wavelet decompositions ⋮ Routes to Chaos in Neural Networks with Random Weights ⋮ Nonlinearity Tests Using the Extrema of a Time Series ⋮ Chaos Structure and Spiral Wave Self-Organization in 2D Coupled Map Lattice ⋮ Characterization of unstable periodic orbits in chaotic attractors and repellers ⋮ LYAPUNOV EXPONENTS OF THE KURAMOTO–SIVASHINSKY PDE ⋮ Machine-aided turbulence theory ⋮ Bifurcations and Hyperchaos in Magnetoconvection of Non-Newtonian Fluids ⋮ Synchronization in chaotic systems ⋮ Learning the tangent space of dynamical instabilities from data ⋮ Deterministic and stochastic bifurcations in two-dimensional electroconvective flows ⋮ ITERATED FUNCTION SYSTEMS WITH PLACE-DEPENDENT PROBABILITIES AND THE INVERSE PROBLEM OF MEASURE APPROXIMATION USING MOMENTS ⋮ Symmetry and chaos in the complex Ginzburg-Landau equation - I. Reflectional symmetries ⋮ Finding Extremal Periodic Orbits with Polynomial Optimization, with Application to a Nine-Mode Model of Shear Flow ⋮ Implementation of an iterative map in the construction of (quasi)periodic instantons: Chaotic aspects and discontinuous rotation numbers ⋮ Lyapunov exponent and criticality in the Hamiltonian mean field model ⋮ Classical dynamics of harmonically trapped interacting particles ⋮ Chaotic attractors from border-collision bifurcations: stable border fixed points and determinant-based Lyapunov exponent bounds ⋮ Artificial sequences and complexity measures ⋮ Dimension of reservoir computers ⋮ Another New Chaotic System: Bifurcation and Chaos Control ⋮ Stabilization of Starobinsky–Vilenkin stochastic inflation by an environmental noise ⋮ Stability, sensitivity and optimisation of chaotic acoustic oscillations ⋮ Characteristics of Chaotic Regimes in a Space-distributed Gyroklystron Model with Delayed Feedback ⋮ A generalized permutation entropy for noisy dynamics and random processes ⋮ ‘Life after death’ in ordinary differential equations with a non-Lipschitz singularity ⋮ Low dimensional manifolds in reservoir computers ⋮ Local Lyapunov exponents and a local estimate of Hausdorff dimension ⋮ Equilibrium states in dynamical systems via geometric measure theory ⋮ Organized structures of two bidirectionally coupled logistic maps ⋮ SYNCHRONIZATION OF SWITCH DYNAMICAL SYSTEMS ⋮ EXTERIOR ALGEBRA-BASED ALGORITHMS TO ESTIMATE LIAPUNOV SPECTRA AND STRETCHING STATISTICS IN HIGH-DIMENSIONAL AND DISTRIBUTED SYSTEMS ⋮ Chaos: From theory to applications for the 80th birthday of Otto E. Rössler ⋮ A dynamical systems model of price bubbles and cycles ⋮ Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics ⋮ Dynamical estimates of chaotic systems from Poincaré recurrences ⋮ Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets ⋮ Abundance of stable periodic behavior in a Red Grouse population model with delay: A consequence of homoclinicity ⋮ Plykin-type attractor in nonautonomous coupled oscillators ⋮ Node-to-node pinning control of complex networks ⋮ A Low-Dimensional Principal Manifold as the "Attractor Backbone" of a Chaotic Beam System ⋮ A large deviation theory-based analysis of heat waves and cold spells in a simplified model of the general circulation of the atmosphere ⋮ Parametric Control to Avoid Bifurcation Based on Maximum Local Lyapunov Exponent ⋮ Stabilization Control of Quasi-periodic Orbits ⋮ Decomposing Signals from Dynamical Systems Using Shadow Manifold Interpolation ⋮ Computational mechanics of molecular systems: Quantifying high-dimensional dynamics by distribution of Poincaré recurrence times ⋮ Existence of physical measures in some excitation–inhibition networks* ⋮ Detecting variation in chaotic attractors ⋮ On a three-dimensional implementation of the baker’s transformation ⋮ OBSERVABILITY OF CHAOS AND CYCLES IN ECOLOGICAL SYSTEMS: LESSONS FROM PREDATOR–PREY MODELS ⋮ GLOBAL DYNAMIC PROPERTIES OF A SYNCHRONOUS MACHINE MODEL ⋮ NOTE ON STRANGE NONFRACTAL ATTRACTORS ⋮ Understanding Chaotic Dynamical Systems ⋮ Successful network inference from time-series data using mutual information rate ⋮ Chaos control of Hastings–Powell model by combining chaotic motions ⋮ Dimension of invariant measures for affine iterated function systems ⋮ Approximating continuous functions on persistence diagrams using template functions ⋮ Convex computation of maximal Lyapunov exponents ⋮ Existence proofs for strange attractors in piecewise-smooth nonlinear Lozi-Hénon and Belykh maps ⋮ Predicting turbulent dynamics with the convolutional autoencoder echo state network ⋮ On the norm equivalence of Lyapunov exponents for regularizing linear evolution equations ⋮ Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flow ⋮ Ergodicity and large deviations in physical systems with stochastic dynamics ⋮ Examining the dynamics of reactive capacity allocation through a chaos Lens ⋮ Generalized Lyapunov exponents and aspects of the theory of deep learning ⋮ Stability analysis of chaotic systems from data ⋮ Chaos measure dynamics in a multifactor model for financial market predictions ⋮ Data based quantification of synchronization ⋮ Wave propagation in one-dimensional nonlinear acoustic metamaterials ⋮ Identifying causally significant features in three-dimensional isotropic turbulence ⋮ Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems ⋮ Attractors. Then and now ⋮ Heterogeneity of the attractor of the Lorenz '96 model: Lyapunov analysis, unstable periodic orbits, and shadowing properties ⋮ Response formulae forn-point correlations in statistical mechanical systems and application to a problem of coarse graining ⋮ An Investigation of the Chaotic Transient for a Boundary Crisis in the Fermi-Ulam Model ⋮ Comparing chaotic and random dynamical systems ⋮ Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations ⋮ RECURRENCE PLOT AND RECURRENCE QUANTIFICATION ANALYSIS TECHNIQUES FOR DETECTING A CRITICAL REGIME. ⋮ Machine failure forewarning via phase-space dissimilarity measures ⋮ Controlled test for predictive power of Lyapunov exponents: Their inability to predict epileptic seizures ⋮ Hyperchaos of arbitrary order generated by a single feedback circuit, and the emergence of chaotic walks ⋮ Flow field topology of transient mixing driven by buoyancy ⋮ On the intrinsic time scales involved in synchronization: A data-driven approach ⋮ Brownian motion and diffusion: From stochastic processes to chaos and beyond ⋮ Multifractal structure of a riddled basin ⋮ Small-scale structure of nonlinearly interacting species advected by chaotic flows ⋮ Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence ⋮ Non-Linear Analysis Approach of Maternal Heart Rate Patterns in Normal and Pre-Eclamptic Pregnancies ⋮ Generalized redundancies for time series analysis ⋮ Relationship of calculating the Jacobian matrices of nonlinear systems and population coding algorithms in neurobiology ⋮ The metric entropy of random dynamical systems in a Banach space: Ruelle inequality ⋮ Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow ⋮ Stabilization of linear time varying systems over uncertain channels ⋮ How often are chaotic transients in spatially extended ecological systems? ⋮ Relationship of calculating the Jacobian matrices of nonlinear systems and population coding algorithms in neurobiology ⋮ An efficient algorithm to determine fractal dimensions of point sets ⋮ The twisted top ⋮ Transition to turbulence in the Reynolds' experiment ⋮ Dynamic and geometric analysis of short time series: a new comparative approach to cell-based biosensors ⋮ Asymptotic orbital stability conditions for flows by estimates of singular values of the linearization ⋮ Nonlinear interaction of four electrostatic waves in a plasma ⋮ Multiscale analysis of economic time series by scale-dependent Lyapunov exponent ⋮ Sum of positive Lyapunov exponents for Kuramoto-Sivashinsky equation ⋮ Incomplete markets and volatility ⋮ Predictability: a way to characterize complexity ⋮ Simple polynomial classes of chaotic jerky dynamics ⋮ Merger and alignment in a reduced model for three-dimensional quasigeostrophic ellipsoidal vortices ⋮ Extraction of nonlinear dynamics from short and noisy time series ⋮ Chaos in mechanical systems -- a review ⋮ Distortions of reconstruction for chaotic attractors ⋮ A superconvergent universality induced by non-associativity ⋮ QUANTIFYING LOCAL INSTABILITY AND PREDICTABILITY OF CHAOTIC DYNAMICAL SYSTEMS BY MEANS OF LOCAL METRIC ENTROPY ⋮ CHAOTIC SYNCHRONIZATION: INFORMATION VIEWPOINT ⋮ DETECTING CHAOTIC DRIVE–RESPONSE GEOMETRY IN GENERALIZED SYNCHRONIZATION ⋮ MODEL SELECTION, CONFIDENCE AND SCALING IN PREDICTING CHAOTIC TIME-SERIES ⋮ STRANGE NONCHAOTIC ATTRACTORS ⋮ ON THE RELATION BETWEEN PREDICTABILITY AND HOMOCLINIC TANGENCIES ⋮ THE COMPUTATION OF LYAPUNOV EXPONENTS FOR PERIODIC TRAJECTORIES ⋮ QUANTUM MAPS WITH SPACE EXTENT: A PARADIGM FOR LATTICE QUANTUM WALKS ⋮ Methods derived from nonlinear dynamics for analysing heart rate variability ⋮ Characteristic Times for the Fermi–Ulam Model ⋮ Notes about the macroscopic fluctuating theory ⋮ Minimal Universal Model for Chaos in Laser with Feedback ⋮ Curry–Yorke route to shearless attractors and coexistence of attractors in dissipative nontwist systems ⋮ Testing for strict stationarity in a random coefficient autoregressive model ⋮ A Method for Estimating the Lyapunov Exponents of Chaotic Time Series Corrupted by Random Noise Using Extended Kalman Filter ⋮ On some aspects of the response to stochastic and deterministic forcings