Analysis of Chaotic Dynamics by the Extended Entropic Chaos Degree

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics.     

Randomly chosen chaotic maps can give rise to nearly ordered behavior

Parrondo’s paradox [J.M.R. Parrondo, G.P. Harmer, D. Abbott, New paradoxical games based on Brownian ratchets, Phys. Rev. Lett. 85 (2000), 5226–5229] (see also [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68–72]) states that two losing gambling games when combined one after the other (either deterministically or randomly) can result in a winning game: that is, a losing game followed by a losing game = a winning game. Inspired by this paradox, a recent study [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] asked an analogous question in discrete time dynamical system: can two chaotic systems give rise to order, namely can they be combined into another dynamical system which does not behave chaotically? Numerical evidence is provided in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] that two chaotic quadratic maps, when composed with each other, create a new dynamical system which has a stable period orbit. The question of what happens in the case of random composition of maps is posed in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124–132] but left unanswered. In this note we present an example of a dynamical system where, at each iteration, a map is chosen in a probabilistic manner from a collection of chaotic maps. The resulting random map is proved to have an infinite absolutely continuous invariant measure (acim) with spikes at two points. From this we show that the dynamics behaves in a nearly ordered manner. When the foregoing maps are applied one after the other, deterministically as in [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68–72], the resulting composed map has a periodic orbit which is stable.     

“Missing moment” and perturbative methods for polynomial iterated function systems

An iterated function system (IFS) over a compact metric space X is defined by a set of contractive maps w_i: X → X, i = 1,…,N, with associated nonzero probabilities p_i > 0, p_i = 1. The “parallel” action of the maps defines a unique compact invariant attractor set A X which supports an invariant measure μ and which is balanced with respect to the p_i. For linear , the invariance of μ yields a relation between the moments g_n = ∫ χⁿ dμ which permits their recursive computation from the initial value g₀ = 1. For nonlinear w_i, however, the moment relations are incomplete and do not permit a recursive computation. This paper describes two methods of obtaining accurate estimates of the moments when the IFS maps w_i are polynomials: (i) application of the necessary Hausdorff conditions on the g_i to obtain convergent upper and lower bounds and (ii) a perturbation expansion approach. The methods are applied to some model problems.     

Time series-based bifurcation diagram reconstruction

We consider the problem of reconstructing bifurcation diagrams (BDs) of maps using time series. This study goes along the same line of ideas presented by Tokunaga et al. [Physica D 79 (1994) 348] and Tokuda et al. [Physica D 95 (1996) 380]. The aim is to reconstruct the BD of a dynamical system without the knowledge of its functional form and its dependence on the parameters. Instead, time series at different parameter values, assumed to be available, are used. A three-layer fully-connected neural network is employed in the approximation of the map. The task of the network is to learn the dynamics of the system as function of the parameters from the available time series. We determine a class of maps for which one can always find a linear subspace in the weight space of the network where the network’s bifurcation structure is qualitatively the same as the bifurcation structure of the map. We discuss a scheme in locating this subspace using the time series. We further discuss how to recognize time series generated by this class of maps. Finally, we propose an algorithm in reconstructing the BDs of this class of maps using predictor functions obtained by neural network. This algorithm is flexible so that other classes of predictors, apart from neural networks, can be used in the reconstruction.     

A subharmonic dynamical bifurcation during in vitro epileptiform activity

Epileptic seizures are considered to result from a sudden change in the synchronization of firing neurons in brain neural networks. We have used an in vitro model of status epilepticus (SE) to characterize dynamical regimes underlying the observed seizure-like activity. Time intervals between spikes or bursts were used as the variable to construct first-return interpeak or interburst interval plots, for studying neuronal population activity during the transition to seizure, as well as within seizures. Return maps constructed for a brief epoch before seizures were used for approximating the local system dynamics during that time window. Analysis of the first-return maps suggests that intermittency is a dynamical regime underlying the observed epileptic activity. This type of analysis may be useful for understanding the collective dynamics of neuronal populations in the normal and pathological brain.     

Chaotic transition of random dynamical systems and chaos synchronization by common noises

We studied the mechanism behind the connection between the transition to chaos of random dynamical systems and the synchronization of chaotic maps driven by external common noises. Near the chaotic transition, the spatial size of random dynamical systems shows an extreme intermittent behavior. By calculating the scaling exponents, we have found that the origin of this intermittent behavior is on-off intermittency. This led us to conclude that chaotic transitions through on-off intermittency can be regarded as a route for random dynamical systems. To clarify this argument, a two-dimensional random dynamical system and two coupled logistic maps driven by external common noises were analyzed.     

Dynamic phase transition from localized to spatiotemporal chaos in coupled circle map with feedback

We investigate coupled circle maps in the presence of feedback and explore various dynamical phases observed in this system of coupled high dimensional maps. We observe an interesting transition from localized chaos to spatiotemporal chaos. We study this transition as a dynamic phase transition. We observe that persistence acts as an excellent quantifier to describe this transition. Taking the location of the fixed point of circle map (which does not change with feedback) as a reference point, we compute a number of sites which have been greater than (less than) the fixed point until time t. Though local dynamics is high dimensional in this case, this definition of persistence which tracks a single variable is an excellent quantifier for this transition. In most cases, we also obtain a well defined persistence exponent at the critical point and observe conventional scaling as seen in second order phase transitions. This indicates that persistence could work as a good order parameter for transitions from fully or partially arrested phase. We also give an explanation of gaps in eigenvalue spectrum of the Jacobian of localized state.     

Return maps of folded nodes and folded saddle-nodes

Folded nodes occur in generic slow-fast dynamical systems with two slow variables. Open regions of initial conditions flow into a folded node in an open set of such systems, so folded nodes are an important feature of generic slow-fast systems. Twisting and linking of trajectories in the vicinity of a folded node have been studied previously, but their consequences for global dynamical behavior have hardly been investigated. One manifestation of the twisting is as "mixed mode oscillations" observed in chemical and neural systems. This paper presents the first systematic numerical study of return maps for trajectories that flow through a region with a folded node. These return maps are approximated by rank-1 maps, and the local twisting of trajectories near a folded node gives rise to multiple turning points in the approximating one dimensional maps. A variant of the forced van der Pol system is used here to illustrate that folded nodes can be a "chaos-generating" mechanism. Folded saddle-nodes occur in generic one-parameter families of slow-fast dynamical systems with two slow variables. These bifurcations give birth to folded nodes. Numerical simulations demonstrate that return maps of systems that are close to a folded saddle-node can be even more complex than those of folded nodes that are far from folded saddles.     

Turning point properties as a method for the characterization of the ergodic dynamics of one-dimensional iterative maps

Dynamical as well as statistical properties of the ergodic and fully developed chaotic dynamics of iterative maps are investigated by means of a turning point analysis. The turning points of a trajectory are hereby defined as the local maxima and minima of the trajectory. An examination of the turning point density directly provides us with the information of the position of the fixed point for the corresponding dynamical system. Dividing the ergodic dynamics into phases consisting of turning points and nonturning points, respectively, elucidates the understanding of the organization of the chaotic dynamics for maps. The turning point map contains information on any iteration of the dynamical law and is shown to possess an asymptotic scaling behaviour which is responsible for the assignment of dynamical structures to the environment of the two fixed points of the map. Universal statistical turning point properties are derived for doubly symmetric maps. Possible applications of the observed turning point properties for the analysis of time series are discussed in some detail. (c) 1997 American Institute of Physics.     

An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.     

Homoclinic tangency and chaotic attractor disappearance in a dripping faucet experiment

A sequence of attractors, reconstructed from interdrops time series data of a leaky faucet experiment, is analyzed as a function of the mean dripping rate. We established the presence of a saddle point and its manifolds in the attractors and we explained the dynamical changes in the system using the evolution of the manifolds of the saddle point, as suggested by the orbits traced in first return maps. The sequence starts at a fixed point and evolves to an invariant torus of increasing diameter (a Hopf bifurcation) that pushes the unstable manifold towards the stable one. The torus breaks up and the system shows a chaotic attractor bounded by the unstable manifold of the saddle. With the attractor expansion the unstable manifold becomes tangential to the stable one, giving rise to the sudden disappearance of the chaotic attractor, which is an experimental observation of a so called chaotic blue sky catastrophe.     

Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. This paper studies discrete data assimilation for the Lorenz equations and the incompressible two-dimensional Navier-Stokes equations. In both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.     

Diffusion,effusion, and chaotic scattering: An exactly solvable Liouvillian dynamics

We study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area-preserving mapping of an infinitely extended strip onto itself. This exactly solvable mapping sustains chaotic behaviors and diffusion processes. The relationship between the diffusion coefficient, the Lyapunov exponent, and the entropy per unit time is derived. The long-lived classical resonances of the Liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation. In this sense, there is a quasi-isomorphism between the resonance spectrum of the Liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation. Furthermore, we show that a fractal repeller is associated to each non-equilibrium state in the isolated and finite multibaker chain. The nonequilibrium states are all unstable with respect to the equilibrium, validating a weak form of the second principle of thermodynamics for the present dynamical system. Consequences of nonequilibrium fractals on classical measurements are discussed. We then describe the open multibaker chain as a scattering system. Fractal properties of chaotic scattering are here shown to be related to diffusion in the chain.     

Algorithmic information for interval maps with an indifferent fixed point and infinite invariant measure

Measuring the average information that is necessary to describe the behavior of a dynamical system leads to a generalization of the Kolmogorov-Sinai entropy. This is particularly interesting when the system has null entropy and the information increases less than linearly with respect to time. We consider a class of maps of the interval with an indifferent fixed point at the origin and an infinite natural invariant measure. We show that the average information that is necessary to describe the behavior of the orbits increases with time n approximately as nalpha, where alpha < 1 depends only on the asymptotic behavior of the map near the origin.     

Electronically-implemented coupled logistic maps

The logistic map is a paradigmatic dynamical system originally conceived to model thediscrete-time demographic growth of a population, which shockingly, shows that discretechaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we designand characterize a simple, low-cost, easy-to-handle, electronic implementation of thelogistic map. In particular, our implementation allows for straightforwardcircuit-modifications to behave as different one-dimensional discrete-time systems. Also,we design a coupling block in order to address the behavior of two coupled maps, although,our design is unrestricted to the discrete-time system implementation and it can begeneralized to handle coupling between many dynamical systems, as in a complex system. Ourfindings show that the isolated and coupled maps’ behavior has a remarkable agreementbetween the experiments and the simulations, even when fine-tuning the parameters with aresolution of ~10^-3. We support these conclusions by comparing the Lyapunovexponents, periodicity of the orbits, and phase portraits of the numerical andexperimental data for a wide range of coupling strengths and map’s parameters.     

Optimal instruments and models for noisy chaos

Analysis of finite, noisy time series data leads to modern statistical inference methods. Here we adapt Bayesian inference for applied symbolic dynamics. We show that reconciling Kolmogorov's maximum-entropy partition with the methods of Bayesian model selection requires the use of two separate optimizations. First, instrument design produces a maximum-entropy symbolic representation of time series data. Second, Bayesian model comparison with a uniform prior selects a minimum-entropy model, with respect to the considered Markov chain orders, of the symbolic data. We illustrate these steps using a binary partition of time series data from the logistic and Henon maps as well as the R?ssler and Lorenz attractors with dynamical noise. In each case we demonstrate the inference of effectively generating partitions and kth-order Markov chain models.     

Spatiotemporal hierarchy of relaxation events, dynamical heterogeneities, and structural reorganization in a supercooled liquid

We identify the pattern of microscopic dynamical relaxation for a two-dimensional glass-forming liquid. On short time scales, bursts of irreversible particle motion, called cage jumps, aggregate into clusters. On larger time scales, clusters aggregate both spatially and temporally into avalanches. This propagation of mobility takes place along the soft regions of the systems, which have been identified by computing isoconfigurational Debye-Waller maps. Our results characterize the way in which dynamical heterogeneity evolves in moderately supercooled liquids and reveal that it is astonishingly similar to the one found for dense glassy granular media.     

Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter

Data assimilation is an iterative approach to the problem of estimating the state of a dynamical system using both current and past observations of the system together with a model for the system’s time evolution. Rather than solving the problem from scratch each time new observations become available, one uses the model to “forecast” the current state, using a prior state estimate (which incorporates information from past data) as the initial condition, then uses current data to correct the prior forecast to a current state estimate. This Bayesian approach is most effective when the uncertainty in both the observations and in the state estimate, as it evolves over time, are accurately quantified. In this article, we describe a practical method for data assimilation in large, spatiotemporally chaotic systems. The method is a type of “ensemble Kalman filter”, in which the state estimate and its approximate uncertainty are represented at any given time by an ensemble of system states. We discuss both the mathematical basis of this approach and its implementation; our primary emphasis is on ease of use and computational speed rather than improving accuracy over previously published approaches to ensemble Kalman filtering. We include some numerical results demonstrating the efficiency and accuracy of our implementation for assimilating real atmospheric data with the global forecast model used by the US National Weather Service.     

One-dimensional strings, random fluctuations, and complex chaotic structures

The effect of random fluctuations on the set of equations resulting from coupling 200 points with one-dimensional maps is studied. This dynamical system exhibits a spatial exponential growth in fluctuations, resulting in a complex chaotic structure. The system suggests a closer look at stability in complex systems.     

Periodic attractors of random truncator maps

This paper introduces the truncator map as a dynamical system on the space of configurations of an interacting particle system. We represent the symbolic dynamics generated by this system as a non-commutative algebra and classify its periodic orbits using properties of endomorphisms of the resulting algebraic structure. A stochastic model is constructed on these endomorphisms, which leads to the classification of the distribution of periodic orbits for random truncator maps. This framework is applied to investigate the periodic transitions of Bornholdt's spin market model.