arXiv subjects

Günter Radons

Publications and source records attributed to Günter Radons.

At least 19 recordsLinked to original sources

arXiv2024

Weak Chaos, Anomalous Diffusion, and Weak Ergodicity Breaking in Systems with Delay

We demonstrate that standard delay systems with a linear instantaneous and a delayed nonlinear term show weak chaos, asymptotically subdiffusive behavior, and weak ergodicity breaking if the nonlinearity is chosen from a specific class of functions. In the limit of large constant delay times, anomalous behavior may not be observable due to exponentially large crossover times. A periodic modulation of the delay causes a strong reduction of the effective dimension of the chaotic phases, leads to hitherto unknown types of solutions, and the occurrence of anomalous diffusion already at short times. The observed anomalous behavior is caused by non-hyperbolic fixed points in function space.

nlin.CD↗

arXiv2023

Laminar chaos in systems with quasiperiodic delay

A new type of chaos called laminar chaos was found in singularly perturbed dynamical systems with periodic time-varying delay [Phys. Rev. Lett. 120, 084102 (2018)]. It is characterized by nearly constant laminar phases, which are periodically interrupted by irregular bursts, where the intensity levels of the laminar phases vary chaotically from phase to phase. In this paper, we demonstrate that laminar chaos can also be observed in systems with quasiperiodic delay, where we generalize the concept of conservative and dissipative delays to such systems. It turns out that the durations of the laminar phases vary quasiperiodically and follow the dynamics of a torus map in contrast to the periodic variation observed for periodic delay. Theoretical and numerical results indicate that introducing a quasiperiodic delay modulation into a time-delay system can lead to a giant reduction of the dimension of the chaotic attractors. By varying the mean delay and keeping other parameters fixed, we found that the Kaplan-Yorke dimension is modulated quasiperiodically over several orders of magnitudes, where the dynamics switches quasiperiodically between different types of high- and low-dimensional types of chaos.

nlin.CD↗

arXiv2023

Pseudo-laminar chaos from on-off intermittency

In finite-dimensional, chaotic, Lorenz-like wave-particle dynamical systems one can find diffusive trajectories, which share their appearance with that of laminar chaotic diffusion [Phys. Rev. Lett. 128, 074101 (2022)] known from delay systems with lag-time modulation. Applying, however, to such systems a test for laminar chaos, as proposed in [Phys. Rev. E 101, 032213 (2020)], these signals fail such test, thus leading to the notion of pseudo-laminar chaos. The latter can be interpreted as integrated periodically driven on-off intermittency. We demonstrate that, on a signal level, true laminar and pseudo-laminar chaos are hardly distinguishable in systems with and without dynamical noise. However, very pronounced differences become apparent when correlations of signals and increments are considered. We compare and contrast these properties of pseudo-laminar chaos with true laminar chaos.

nlin.CD↗

arXiv2022

Suppression of Quasiperiodicity in Circle Maps with Quenched Disorder

We show that introducing quenched disorder into a circle map leads to the suppression of quasiperiodic behavior in the limit of large system sizes. Specifically, for most parameters the fraction of disorder realizations showing quasiperiodicity decreases with the system size and eventually vanishes in the limit of infinite size, where almost all realizations show mode-locking. Consequently, in this limit, and in strong contrast to standard circle maps, almost the whole parameter space corresponding to invertible dynamics consists of Arnold tongues.

nlin.CD↗

arXiv2022

Infinite ergodic theory for three heterogeneous stochastic models with application to subrecoil laser cooling

We compare ergodic properties of the kinetic energy for three stochastic models of subrecoil-laser-cooled gases. One model is based on a heterogeneous random walk (HRW), another is an HRW with long-range jumps (the exponential model), and the other is a mean-field-like approximation of the exponential model (the deterministic model). All the models show an accumulation of the momentum at zero in the long-time limit, and a formal steady state cannot be normalized, i.e., there exists an infinite invariant density. We obtain the exact form of the infinite invariant density and the scaling function for the exponential and deterministic models and devise a useful approximation for the momentum distribution in the HRW model. While the models are kinetically non-identical, it is natural to wonder whether their ergodic properties share common traits, given that they are all described by an infinite invariant density. We show that the answer to this question depends on the type of observable under study. If the observable is integrable, the ergodic properties such as the statistical behavior of the time averages are universal as they are described by the Darling-Kac theorem. In contrast, for non-integrable observables, the models in general exhibit non-identical statistical laws. This implies that focusing on non-integrable observables, we discover non-universal features of the cooling process, that hopefully can lead to a better understanding of the particular model most suitable for a statistical description of the process. This result is expected to hold true for many other systems, beyond laser cooling.

cond-mat.stat-mech↗

arXiv2022

Chaotic Diffusion in Delay Systems: Giant Enhancement by Time Lag Modulation

We consider a typical class of systems with delayed nonlinearity, which we show to exhibit chaotic diffusion. It is demonstrated that a periodic modulation of the time-lag can lead to an enhancement of the diffusion constant by several orders of magnitude. This effect is the largest if the circle map defined by the modulation shows mode locking and more specifically, fulfills the conditions for laminar chaos. Thus we establish for the first time a connection between Arnold tongue structures in parameter space and diffusive properties of a delay system. Counterintuitively, the enhancement of diffusion is accompanied by a strong reduction of the effective dimensionality of the system.

nlin.CD↗

arXiv2022

Nonergodicity of $d$-dimensional generalized Lévy walks and their relation to other space-time coupled models

We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent Letter [Phys. Rev. Lett. 120, 104501 (2018)], give profound interpretations, and especially emphasize three surprising results: First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-averaged squared displacements show superdiffusion. Third, we recognize that the fluctuations of the time-averaged squared displacements can become so large that the ergodicity breaking parameter diverges, what we call infinitely strong ergodicity breaking. The latter phenomenon can also occur for paramter values where the lag-time dependence of the mean-squared displacements is linear indicating normal diffusion. In order to numerically determine the full distribution of time-averaged squared displacements, we use importance sampling. For an embedding of our new findings into existing results in the literature, we define a more general model which we call variable speed generalized Lévy walk and which includes well known models from the literature as special cases such as the space-time coupled Lévy flight or the anomalous Drude model. We discuss and interpret our findings regarding the generalized Lévy walk in detail and compare them with the nonergodicity of the other space-time coupled models following from the more general model.

cond-mat.stat-mech↗

arXiv2022

Laminar Chaos

We show that the output of systems with time-varying delay can exhibit a new kind of chaotic behavior characterized by laminar phases, which are periodically interrupted by irregular bursts. Within each laminar phase the output intensity remains almost constant, but its level varies chaotically from phase to phase. In scalar systems the periodic dynamics of the lengths and the chaotic dynamics of the intensity levels can be understood and also tuned via two one-dimensional maps, which can be deduced from the nonlinearity of the delay equation and from the delay variation, respectively.

nlin.CD↗

arXiv2022

Anti-persistent random walks in time-delayed systems

We show that the occurrence of chaotic diffusion in a typical class of time-delayed systems with linear instantaneous and nonlinear delayed term can be well described by an anti-persistent random walk. We numerically investigate the dependence of all relevant quantities characterizing the random walk on the strength of the nonlinearity and on the delay. With the help of analytical considerations, we show that for a decreasing nonlinearity parameter the resulting dependence of the diffusion coefficient is well described by Markov processes of increasing order.

cond-mat.stat-mech↗

arXiv2022

Chaotic Diffusion of Dissipative Solitons: From Anti-Persistent Random Walk to Hidden Markov Models

In previous publications, we showed that the incremental process of the chaotic diffusion of dissipative solitons in a prototypical complex Ginzburg-Landau equation, known, e.g., from nonlinear optics, is governed by a simple Markov process leading to an Anti-Persistent Random Walk of motion or by a more complex Hidden Markov Model with continuous output densities. In this article, we reveal the transition between these two models by studying the soliton dynamics in dependence on the main bifurcation parameter of the Ginzburg-Landau equation and identify the underlying hidden Markov processes. These models capture the non-trivial decay of correlations in jump widths and symbol sequences representing the soliton motion, the statistics of anti-persistent walk episodes, and the multimodal density of the jump widths. We demonstrate that there exists a physically meaningful reduction of the dynamics of an infinite-dimensional deterministic system to one of a probabilistic finite state machine and provide a deeper understanding of the soliton dynamics under parameter variation of the underlying nonlinear dynamics.

nlin.CD↗

arXiv2021

The dynamics of a driven harmonic oscillator coupled to pairwise interacting Ising spins in random fields

In general we are interested in dynamical systems coupled to complex hysteresis. Therefore as a first step we investigated recently the dynamics of a periodically driven damped harmonic oscillator coupled to independent Ising spins in a random field. Although such a system does not produce hysteresis, we showed how to characterize the dynamics of such a piecewise-smooth system, especially in the case of a large number of spins [P. Zech, A. Otto, and G. Radons, Phys. Rev. E 101, 042217 (2020)]. In this paper we extend our model to spin dimers, thus pairwise interacting spins. We show in which cases two interacting spins can show elementary hysteresis and we give a connection to the Preisach model, which allows us to consider an infinite number of spin-pairs. This thermodynamic limit leads us to a dynamical system with an additional hysteretic force in the form of a generalized play operator. By using methods from general chaos theory, piecewise-smooth system theory and statistics we investigate the chaotic behavior of the dynamical system for a few spins and also in case of a larger number of spins by calculating bifurcation diagrams, Lyapunov exponents, fractal dimensions and self-averaging properties. We find that the fractal dimensions and the magnetization are in general not self-averaging quantities. We show, how the dynamical properties of the piecewise-smooth system for a large number of spins differs from the system in its thermodynamic limit.

nlin.CD↗

arXiv2021

Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory

The velocity distribution of a classical gas of atoms in thermal equilibrium is the normal Maxwell distribution. It is well known that for sub-recoiled laser cooled atoms Lévy statistics and deviations from usual ergodic behaviour come into play. Here we show how tools from infinite ergodic theory describe the cool gas. Specifically, we derive the scaling function and the infinite invariant density of a stochastic model for the momentum of laser cooled atoms using two approaches. The first is a direct analysis of the master equation and the second following the analysis of Bertin and Bardou using the lifetime dynamics. The two methods are shown to be identical, but yield different insights into the problem. In the main part of the paper we focus on the case where the laser trapping is strong, namely the rate of escape from the velocity trap is $R(v) \propto |v|^α$ for $v \to 0$ and $α>1$. We construct a machinery to investigate the time averages of physical observables and their relation to ensemble averages. The time averages are given in terms of functionals of the individual stochastic paths, and here we use a generalisation of Lévy walks to investigate the ergodic properties of the system. Exploring the energy of the system, we show that when $α=3$ it exhibits a transition between phases where it is either an integrable or non integrable observable, with respect to the infinite invariant measure. This transition corresponds to very different properties of the mean energy, and to a discontinuous behaviour of the fluctuations. Since previous experimental work showed that both $α=2$ and $α=4$ are attainable we believe that both phases could be explored also experimentally.

cond-mat.stat-mech↗

arXiv2021

Transitions in the ergodicity of subrecoil-laser-cooled gases

With subrecoil-laser-cooled atoms one may reach nano-Kelvin temperatures while the ergodic properties of these systems do not follow usual statistical laws. Instead, due to an ingenious trapping mechanism in momentum space, power-law-distributed sojourn times are found for the cooled particles. Here, we show how this gives rise to a statistical-mechanical framework based on infinite ergodic theory, which replaces ordinary ergodic statistical physics of a thermal gas of atoms. In particular, the energy of the system exhibits a sharp discontinuous transition in its ergodic properties. Physically this is controlled by the fluorescence rate, but more profoundly it is a manifestation of a transition for any observable, from being an integrable to becoming a non-integrable observable, with respect to the infinite (non-normalised) invariant density.

cond-mat.stat-mech↗

arXiv2020

Infinite invariant density in a semi-Markov process with continuous state variables

We report on a fundamental role of a non-normalized formal steady state, i.e., an infinite invariant density, in a semi-Markov process where the state is determined by the inter-event time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized Lévy walk model and the energy of a particle in the trap model. In our model, the inter-event-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long inter-event times imply small state values. We find two scaling laws describing the density for the state value, which accumulates in the vicinity of zero in the long-time limit. These laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density. Moreover, we provide two distributional limit theorems for time-averaged observables in these non-stationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages.

cond-mat.stat-mech↗

arXiv2020

The dynamics of a driven harmonic oscillator coupled to independent Ising spins in random fields

We aim at an understanding of the dynamical properties of a periodically driven damped harmonic oscillator coupled to a Random Field Ising Model (RFIM) at zero temperature, which is capable to show complex hysteresis. The system is a combination of a continuous (harmonic oscillator) and a discrete (RFIM) subsystem, which classifies it as a hybrid system. In this paper we focus on the hybrid nature of the system and consider only independent spins in quenched random local fields, which can already lead to complex dynamics such as chaos and multistability. We study the dynamic behavior of this system by using the theory of piecewise-smooth dynamical systems and discontinuity mappings. Specifically, we present bifurcation diagrams, Lyapunov exponents as well as results for the shape and the dimensions of the attractors and the self-averaging behavior of the attractor dimensions and the magnetization. Furthermore we investigate the dynamical behavior of the system for an increasing number of spins and the transition to the thermodynamic limit, where the system behaves like a driven harmonic oscillator with an additional nonlinear smooth external force.

nlin.CD↗

arXiv2020

Laminar Chaos in experiments and nonlinear delayed Langevin equations: A time series analysis toolbox for the detection of Laminar Chaos

Recently, it was shown that certain systems with large time-varying delay exhibit different types of chaos, which are related to two types of time-varying delay: conservative and dissipative delays. The known high-dimensional Turbulent Chaos is characterized by strong fluctuations. In contrast, the recently discovered low-dimensional Laminar Chaos is characterized by nearly constant laminar phases with periodic durations and a chaotic variation of the intensity from phase to phase. In this paper we extend our results from our preceding publication [J. D. Hart, R. Roy, D. Müller-Bender, A. Otto, and G. Radons, PRL 123 154101 (2019)], where it is demonstrated that Laminar Chaos is a robust phenomenon, which can be observed in experimental systems. We provide a time-series analysis toolbox for the detection of robust features of Laminar Chaos. We benchmark our toolbox by experimental time series and time series of a model system which is described by a nonlinear Langevin equation with time-varying delay. The benchmark is done for different noise strengths for both the experimental system and the model system, where Laminar Chaos can be detected, even if it is hard to distinguish from Turbulent Chaos by a visual analysis of the trajectory.

nlin.AO↗

arXiv2018

Normal and anomalous random walks of 2-d solitons

Solitons, which describe the propagation of concentrated beams of light through nonlinear media, can exhibit a variety of behaviors as a result of the intrinsic dissipation, diffraction, and the nonlinear effects. One of these phenomena, modeled by the complex Ginzburg-Landau equation, are chaotic explosions, transient enlargements of the soliton that may induce random transversal displacements, which in the long run lead to a random walk of the soliton center. As we show in this work, the transition from non-moving to moving solitons is not a simple bifurcation but includes a sequence of normal and anomalous random walks. We analyze their statistics with the distribution of generalized diffusivities, a novel approach that has been used successfully for characterizing anomalous diffusion.

nlin.PS↗

arXiv2018

Anomalous diffusion of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions

We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a `simple' and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions it becomes characterized by normal diffusion.

nlin.PS↗