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H. Kantz

Publications and source records attributed to H. Kantz.

At least 19 recordsLinked to original sources
arXiv2023

Complexity and transition to chaos in coupled Adler-type oscillators

Coupled non-linear 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 non-trivial 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.

nlin.AO↗
arXiv2018

Large deviations of time-averaged statistics for Gaussian processes

In this paper we study the large deviations of time averaged mean square displacement (TAMSD) for Gaussian processes. The theory of large deviations is related to the exponential decay of probabilities of large fluctuations in random systems. From the mathematical point of view a given statistics satisfies the large deviation principle, if the probability that it belongs to a certain range decreases exponentially. The TAMSD is one of the main statistics used in the problem of anomalous diffusion detection. Applying the theory of generalized chi-squared distribution and sub-gamma random variables we prove the upper bound for large deviations of TAMSD for Gaussian processes. As a special case we consider fractional Brownian motion, one of the most popular models of anomalous diffusion. Moreover, we derive the upper bound for large deviations of the estimator for the anomalous diffusion exponent.

math.PR↗
arXiv2018

Probabilistic properties of detrended fluctuation analysis for Gaussian processes

The detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range correlations in time series. Although DFA has found many interesting applications and has been shown as one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper we study probabilistic properties of DFA for Gaussian processes. The main attention is paid to the distribution of the squared error sum of the detrended process. This allows us to find the expected value and the variance of the fluctuation function of DFA for a Gaussian process of general form. The results obtained can serve as a starting point for analyzing the statistical properties of the DFA-based estimators for the fluctuation and correlation parameters. The obtained theoretical formulas are supported by numerical simulations of particular Gaussian processes possessing short-and long-memory behaviour.

cond-mat.stat-mech↗
arXiv2013

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 walk 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, by 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.

cond-mat.stat-mech↗
arXiv2011

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 non-degenerate 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.

cond-mat.stat-mech↗
arXiv2011

Predicting Failures of Point Forecasts

The predictability of errors in deterministic temperature forecasts is investigated. More precisely, the aim is to issue warnings whenever the differences between forecast and verification exceed a given threshold. The warnings are generated by analyzing the output of an ensemble forecast system in terms of a decision making approach. The quality of the resulting predictions is evaluated by computing receiver operating characteristics, the Brier score, and the Ignorance score. Special emphasis is also given to the question whether rare events are better predictable.

physics.ao-ph↗
arXiv2011

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.

cond-mat.stat-mech↗
arXiv2011

Probability distribution function for systems driven by superheavy-tailed noise

We develop a general approach for studying the cumulative probability distribution function of localized objects (particles) whose dynamics is governed by the first-order Langevin equation driven by superheavy-tailed noise. Solving the corresponding Fokker-Planck equation, we show that due to this noise the distribution function can be divided into two different parts describing the surviving and absorbing states of particles. These states and the role of superheavy-tailed noise are studied in detail using the theory of slowly varying functions.

cond-mat.stat-mech↗
arXiv2010

Continuous-time random walk theory of superslow diffusion

Superslow diffusion, i.e., the long-time diffusion of particles whose mean-square displacement (variance) grows slower than any power of time, is studied in the framework of the decoupled continuous-time random walk model. We show that this behavior of the variance occurs when the complementary cumulative distribution function of waiting times is asymptotically described by a slowly varying function. In this case, we derive a general representation of the laws of superslow diffusion for both biased and unbiased versions of the model and, to illustrate the obtained results, consider two particular classes of waiting-time distributions.

cond-mat.stat-mech↗
arXiv2010

Biased diffusion in a piecewise linear random potential

We study the biased diffusion of particles moving in one direction under the action of a constant force in the presence of a piecewise linear random potential. Using the overdamped equation of motion, we represent the first and second moments of the particle position as inverse Laplace transforms. By applying to these transforms the ordinary and the modified Tauberian theorem, we determine the short- and long-time behavior of the mean-square displacement of particles. Our results show that while at short times the biased diffusion is always ballistic, at long times it can be either normal or anomalous. We formulate the conditions for normal and anomalous behavior and derive the laws of biased diffusion in both these cases.

cond-mat.stat-mech↗
arXiv2010

Langevin equation with super-heavy-tailed noise

We extend the Langevin approach to a class of driving noises whose generating processes have independent increments with super-heavy-tailed distributions. The time-dependent generalized Fokker-Planck equation that corresponds to the first-order Langevin equation driven by such a noise is derived and solved exactly. This noise generates two probabilistic states of the system, survived and absorbed, that are equivalent to those for a classical particle in an absorbing medium. The connection between the rate of absorption and the super-heavy-tailed distribution of the increments is established analytically. A numerical scheme for the simulation of the Langevin equation with super-heavy-tailed noise is developed and used to verify our theoretical results.

cond-mat.stat-mech↗
arXiv2010

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.

cond-mat.stat-mech↗
arXiv2009

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.

cond-mat.dis-nn↗
arXiv2008

Statistical analysis of wind speed fluctuation and increments of non-stationary atmospheric boundary layer turbulence

We study the statistics of the horizontal component of atmospheric boundary layer wind speed. Motivated by its non-stationarity, we investigate which parameters remain constant or can be regarded as being piece-wise constant and explain how to estimate them. We will verify the picture of natural atmospheric boundary layer turbulence to be composed of successively occurring close to ideal turbulence with different parameters. The first focus is put on the fluctuation of wind speed around its mean behaviour. We describe a method estimating the proportionality factor between the standard deviation of the fluctuation and the mean wind speed and analyse its time dependence. The second focus is put on the wind speed increments. We investigate the increment distribution and use an algorithm based on superstatistics to quantify the time dependence of the parameters describing the distribution. Applying the introduced tools yields a comprehensive description of the wind speed in the atmospheric boundary layer.

physics.data-an↗
arXiv2008

When are Extreme Events the better predictable, the larger they are?

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 magnitude 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 magnitude. 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.

physics.data-an↗
arXiv2007

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 {\it exist} and {\it 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.

nlin.CD↗
arXiv2006

Hoelder-exponent-MFDFA-based test for long-range correlations in pseudorandom sequences

We discuss the problem for detecting long-range correlations in sequences of values obtained by generators of pseudo-random numbers. The basic idea is that the H{ö}lder exponent for a sufficiently long sequence of uncorrelated random numbers has the fixed value $h=1/2$. The presence of long-range correlations leads to deviation from this value. We calculate the H{ö}lder exponent by the method of multifractal detrended fluctuation analysis (MFDFA). We discuss frequently used tests for randomness, finite sample properties of the MFDFA, and the conditions for a correct application of the method. We observe that the fluctuation function $F_{q}$ used in the MFDFA reacts to trends caused by low periodicity presented in the pseudo-random number generator. In order to select appropriate generators from the numerous programs we propose a test for the ensemble properties of the generated pseudo-random sequences with respect to their robustness against presence of long-range correlations, and a selection rule which orders the generators that pass the test. Selecting generators that successfully pass the ensemble test and have good performance with respect to the selection rule is not enough. For the selected generator we have to choose appropriate pseudo-random sequences for the length of the sequence required by the solved problem. This choice is based on the closeness of the H{ö}lder exponent of the generated sequence to its value 1/2 characteristic for the case of absence of correlations.

physics.data-an↗