Complexity and transition to chaos in coupled Adler-type oscillators.

Coupled nonlinear oscillators are ubiquitous in dynamical studies. A wealth of behaviors have been found mostly for globally coupled systems. From a complexity perspective, less studied have been systems with local coupling, which is the subject of this contribution. The phase approximation is used, as weak coupling is assumed. In particular, the so-called needle region, in parameter space, for Adler-type oscillators with nearest neighbors coupling is carefully characterized. The reason for this emphasis is that, in the border of this region to the surrounding chaotic one, computation enhancement at the edge of chaos has been reported. The present study shows that different behaviors within the needle region can be found and a smooth change of dynamics could be identified. Entropic measures further emphasize the region's heterogeneous nature with interesting features, as seen in the spatiotemporal diagrams. The occurrence of wave-like patterns in the spatiotemporal diagrams points to nontrivial correlations in both dimensions. The wave patterns change as the control parameters change without exiting the needle region. Spatial correlation is only achieved locally at the onset of chaos, with different clusters of oscillators behaving coherently while disordered boundaries appear between them.

Limiting distributions of continuous-time random walks with superheavy-tailed waiting times.

We study the long-time behavior of the scaled walker (particle) position associated with decoupled continuous-time random walks which is characterized by superheavy-tailed distribution of waiting times and asymmetric heavy-tailed distribution of jump lengths. Both the scaling function and the corresponding limiting probability density are determined for all admissible values of tail indexes describing the jump distribution. To analytically investigate the limiting density function, we derive a number of different representations of this function and, in this way, establish its main properties. We also develop an efficient numerical method for computing the limiting probability density and compare our analytical and numerical results.

Continuous-time random walk with a superheavy-tailed distribution of waiting times.

We study the long-time behavior of the probability density associated with the decoupled continuous-time random walk which is characterized by a superheavy-tailed distribution of waiting times. It is shown that, if the random walk is unbiased (biased) and the jump distribution has a finite second moment, then the properly scaled probability density converges in the long-time limit to a symmetric two-sided (an asymmetric one-sided) exponential density. The convergence occurs in such a way that all the moments of the probability density grow slower than any power of time. As a consequence, the reference random walk can be viewed as a generic model of superslow diffusion. A few examples of superheavy-tailed distributions of waiting times that give rise to qualitatively different laws of superslow diffusion are considered.

Asymptotic solutions of decoupled continuous-time random walks with superheavy-tailed waiting time and heavy-tailed jump length distributions.

We study the long-time behavior of decoupled continuous-time random walks characterized by superheavy-tailed distributions of waiting times and symmetric heavy-tailed distributions of jump lengths. Our main quantity of interest is the limiting probability density of the position of the walker multiplied by a scaling function of time. We show that the probability density of the scaled walker position converges in the long-time limit to a nondegenerate one only if the scaling function behaves in a certain way. This function as well as the limiting probability density are determined in explicit form. Also, we express the limiting probability density which has heavy tails in terms of the Fox H function and find its behavior for small and large distances.

Anomalous biased diffusion in a randomly layered medium.

We present analytical results for the biased diffusion of particles moving under a constant force in a randomly layered medium. The influence of this medium on the particle dynamics is modeled by a piecewise constant random force. The long-time behavior of the particle position is studied in the frame of a continuous-time random walk on a semi-infinite one-dimensional lattice. We formulate the conditions for anomalous diffusion, derive the diffusion laws, and analyze their dependence on the particle mass and the distribution of the random force.

Directed transport in periodically rocked random sawtooth potentials.

We study directed transport of overdamped particles in a periodically rocked random sawtooth potential. Two transport regimes can be identified which are characterized by a nonzero value of the average velocity of particles and a zero value, respectively. The properties of directed transport in these regimes are investigated both analytically and numerically in terms of a random sawtooth potential and a periodically varying driving force. Precise conditions for the occurrence of transition between these two transport regimes are derived and analyzed in detail.

Influence of the event magnitude on the predictability of an extreme event.

We investigate the predictability of extreme events in time series. The focus of this work is to understand under which circumstances large events are better predictable than smaller events. Therefore we use a simple prediction algorithm based on precursory structures which are identified using the maximum likelihood principle. Using the receiver operator characteristic curve as a measure for the quality of predictions we find that the dependence on the event size is closely linked to the probability distribution function of the underlying stochastic process. We evaluate this dependence on the probability distribution function analytically and numerically. If we assume that the optimal precursory structures are used to make the predictions, we find that large increments are better predictable if the underlying stochastic process has a Gaussian probability distribution function, whereas larger increments are harder to predict if the underlying probability distribution function has a power-law tail. In the case of an exponential distribution function we find no significant dependence on the event size. Furthermore we compare these results with predictions of increments in correlated data, namely, velocity increments of a free jet flow. The velocity increments in the free jet flow are in dependence on the time scale either asymptotically Gaussian or asymptotically exponential distributed. The numerical results for predictions within free jet data are in good agreement with the previous analytical considerations for random numbers.

Prevalence of marginally unstable periodic orbits in chaotic billiards.

The dynamics of chaotic billiards is significantly influenced by coexisting regions of regular motion. Here we investigate the prevalence of a different fundamental structure, which is formed by marginally unstable periodic orbits and stands apart from the regular regions. We show that these structures both exist and strongly influence the dynamics of locally perturbed billiards, which include a large class of widely studied systems. We demonstrate the impact of these structures in the quantum regime using microwave experiments in annular billiards.

Direction of coupling from phases of interacting oscillators: a permutation information approach.

We introduce a directionality index for a time series based on a comparison of neighboring values. It can distinguish unidirectional from bidirectional coupling, as well as reveal and quantify asymmetry in bidirectional coupling. It is tested on a numerical model of coupled van der Pol oscillators, and applied to cardiorespiratory data from healthy subjects. There is no need for preprocessing and fine-tuning the parameters, which makes the method very simple, computationally fast and robust.

Curved structures in recurrence plots: the role of the sampling time.

We give a theoretical explanation of the formation of the curved macropatterns in the recurrence plots of sinusoidal signals, with nonstationarity in the phase or in the frequency. We show that the large time scales observed and the curved structures are the artificial product of the discretization of the signal. Recurrence plots are highly sensitive to the phase error introduced by the sampling, and we show that this characteristic can be used to detect very small (approximately 0.5%) phase or frequency shifts of the carrier frequency.

Recurrence plot analysis of nonstationary data: the understanding of curved patterns.

Recurrence plots of the calls of the Nomascus concolor (Western black crested gibbon) and Hylobates lar (White-handed gibbon) show characteristic circular, curved, and hyperbolic patterns superimposed to the main temporal scale of the signal. It is shown that these patterns are related to particular nonstationarities in the signal. Some of them can be reproduced by artificial signals like frequency modulated sinusoids and sinusoids with time divergent frequency. These modulations are too faint to be resolved by conventional time-frequency analysis with similar precision. Therefore, recurrence plots act as a magnifying glass for the detection of multiple temporal scales in slightly modulated signals. The detected phenomena in these acoustic signals can be explained in the biomechanical context by taking in account the role of the muscles controlling the vocal folds.

Invariant densities of delayed maps in the limit of large time delay.

The marginal invariant density of chaotic attractors of scalar systems with time delayed feedback has an asymptotic form in the limit of large delay. It is well known that the dimension and the entropy of such attractors obey interesting scaling laws in this limit, but very little has been said about properties of the invariant density. We present general considerations, detailed analytical results in low order perturbation theory for a particular model, and numerics for understanding the asymptotic behavior of the projections of the invariant density. Our approach clarifies how the analytical properties of the model determine the behavior of the marginal invariant densities for large delay times.

Indispensable finite time corrections for Fokker-Planck equations from time series data.

The reconstruction of Fokker-Planck equations from observed time series data suffers strongly from finite sampling rates. We show that previously published results are degraded considerably by such effects. We present correction terms which yield a robust estimation of the diffusion terms, together with a novel method for one-dimensional problems. We apply these methods to time series data of local surface wind velocities, where the dependence of the diffusion constant on the state variable shows a different behavior than previously suggested.

Reconstruction of the parameter spaces of dynamical systems.

Parameter variations in the equations of motion of dynamical systems are identified by time series analysis. The information contained in time series data is transformed and compressed to feature vectors. The space of feature vectors is an embedding for the unobserved parameters of the system. We show that the smooth variation of d system parameters can lead to paths of feature vectors on smooth d-dimensional manifolds in feature space, provided the latter is high-dimensional enough. The number of varying parameters and the nature of their variation can thus be identified. The method is illustrated using numerically generated data and experimental data from electromotors. Complications arising from bifurcations in deterministic dynamical systems are shown to disappear for slightly noisy systems.

Periodic orbits and topological entropy of delayed maps.

The periodic orbits of a nonlinear dynamical system provide valuable insight into the topological and metric properties of its chaotic attractors. In this paper we describe general properties of periodic orbits of dynamical systems with feedback delay. In the case of delayed maps, these properties enable us to provide general arguments about the boundedness of the topological entropy in the high delay limit. As a consequence, all the metric entropies can be shown to be bounded in this limit. The general considerations are illustrated in the cases of Bernoulli-like and Hénon-like delayed maps.

Relation between coupled map lattices and kinetic ising models

A spatially one-dimensional coupled map lattice possessing the same symmetries as the Miller-Huse model is introduced. Our model is studied analytically by means of a formal perturbation expansion which uses weak coupling and the vicinity to a symmetry breaking bifurcation point. In parameter space four phases with different ergodic behavior are observed. Although the coupling in the map lattice is diffusive, antiferromagnetic ordering is predominant. Via coarse graining the deterministic model is mapped to a master equation which establishes an equivalence between our system and a kinetic Ising model. Such an approach sheds some light on the dependence of the transient behavior on the system size and the nature of the phase transitions.

Chaos or noise: difficulties of a distinction

In experiments, the dynamical behavior of systems is reflected in time series. Due to the finiteness of the observational data set, it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension. These restrictions limit our ability to distinguish between signals generated by different systems, such as regular, chaotic, or stochastic ones, when analyzed from a time series point of view. We propose to classify the signal behavior, without referring to any specific model, as stochastic or deterministic on a certain scale of the resolution epsilon, according to the dependence of the (epsilon,tau) entropy, h(epsilon, tau), and the finite size Lyapunov exponent lambda(epsilon) on epsilon.

Analysis of vocal disorders in a feature space.

This paper provides a way to classify vocal disorders for clinical applications. This goal is achieved by means of geometric signal separation in a feature space. Typical quantities from chaos theory (like entropy, correlation dimension and first lyapunov exponent) and some conventional ones (like autocorrelation and spectral factor) are analysed and evaluated, in order to provide entries for the feature vectors. A way of quantifying the amount of disorder is proposed by means of a healthy index that measures the distance of a voice sample from the centre of mass of both healthy and sick clusters in the feature space. A successful application of the geometrical signal separation is reported, concerning distinction between normal and disordered phonation.

Denoising human speech signals using chaoslike features.

A local projective noise reduction scheme, originally developed for low-dimensional stationary deterministic chaotic signals, is successfully applied to human speech. This is possible by exploiting properties of the speech signal which resemble structure exhibited by deterministic dynamical systems. In high-dimensional embedding spaces, the strong inherent nonstationarity is resolved as a sequence of many different dynamical regimes of moderate complexity.

Local estimates for entropy densities in coupled map lattices

We present a method to derive an upper bound for the entropy density of coupled map lattices with local interactions from local observations. To do this, we use an embedding technique that is a combination of time delay and spatial embedding. This embedding allows us to identify the local character of the equations of motion. Based on this method we present an approximate estimate of the entropy density by the correlation integral.