Günter Radons

Hyperbolicity and the Effective Dimension of Spatially Extended Dissipative Systems

Using covariant Lyapunov vectors, we reveal a split of the tangent space of standard models of one-dimensional dissipative spatiotemporal chaos: A finite extensive set of N dynamically entangled vectors with frequent common tangencies describes all of the physically relevant dynamics and is hyperbolically separated from possibly infinitely many isolated modes representing trivial, exponentially decaying perturbations. We argue that N can be interpreted as the number of effective degrees of freedom, which has to be taken into account in numerical integration and control issues.

Subdiffusive continuous time random walks and weak ergodicity breaking analyzed with the distribution of generalized diffusivities

We propose a new tool for analyzing data from anomalous diffusion processes: The distribution of generalized diffusivities pα(D,τ) describes the fluctuations during the diffusion process around the generalized diffusion coefficient obtained from the mean squared displacement and its τ-dependence captures the non-trivial part of the process dynamics. We apply this tool to subdiffusive continuous time random walks which are known to show weak ergodicity breaking. We characterize how the distribution of generalized diffusivities obtained from an ensemble of trajectories differs from the distribution obtained as a time average from one single-particle trajectory and show how such an analysis leads to a deeper understanding of weak ergodicity breaking.

The problem of spurious Lyapunov exponents in time series analysis and its solution by covariant Lyapunov vectors

We briefly recall the methods for the determination of Lyapunov exponents from time series data and their limitations. One particular problem is given by the fact that reconstructed phase spaces have usually extra dimensions compared to the true phase space of a dynamical system, leading to extra, so called spurious Lyapunov exponents. Several methods to identify the true ones have been proposed which do not give satisfactory results. We show that the geometric information contained in covariant Lyapunov vectors can be used to identify the true exponents. We illustrate its use and its limitations by applying it to experimental NMR laser data.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Lyapunov analysis: from dynamical systems theory to applications?.

Characterizing N-dimensional anisotropic Brownian motion by the distribution of diffusivities

Anisotropic diffusion processes emerge in various fields such as transport in biological tissue and diffusion in liquid crystals. In such systems, the motion is described by a diffusion tensor. For a proper characterization of processes with more than one diffusion coefficient, an average description by the mean squared displacement is often not sufficient. Hence, in this paper, we use the distribution of diffusivities to study diffusion in a homogeneous anisotropic environment. We derive analytical expressions of the distribution and relate its properties to an anisotropy measure based on the mean diffusivity and the asymptotic decay of the distribution. Both quantities are easy to determine from experimental data and reveal the existence of more than one diffusion coefficient, which allows the distinction between isotropic and anisotropic processes. We further discuss the influence on the analysis of projected trajectories, which are typically accessible in experiments. For the experimentally most relevant cases of two- and three-dimensional anisotropic diffusion, we derive specific expressions, determine the diffusion tensor, characterize the anisotropy, and demonstrate the applicability for simulated trajectories.

Lyapunov modes in extended systems

Hydrodynamic Lyapunov modes, which have recently been observed in many extended systems with translational symmetry, such as hard sphere systems, dynamic XY models or Lennard–Jones fluids, are nowadays regarded as fundamental objects connecting nonlinear dynamics and statistical physics. We review here our recent results on Lyapunov modes in extended system. The solution to one of the puzzles, the appearance of good and ‘vague’ modes, is presented for the model system of coupled map lattices. The structural properties of these modes are related to the phase space geometry, especially the angles between Oseledec subspaces, and to fluctuations of local Lyapunov exponents. In this context, we report also on the possible appearance of branches splitting in the Lyapunov spectra of diatomic systems, similar to acoustic and optical branches for phonons. The final part is devoted to the hyperbolicity of partial differential equations and the effective degrees of freedom of such infinite-dimensional systems.

Analytical investigation of modulated time-delayed feedback control

The influence of time-dependent control parameters on time-delayed feedback schemes for control of chaos is investigated by analytical means. For the logistic map the linear stability of the period-two orbit subjected to a modulated time-delayed feedback loop is calculated. We find enhanced control performance due to phase lags between the periodic orbit and the controller.

Decohering localized waves.

In the absence of confinement, localization of waves takes place due to randomness or nonlinearity and relies on their phase coherence. We quantitatively probe the sensitivity of localized wave packets to random phase fluctuations and confirm the necessity of phase coherence for localization. Decoherence resulting from a dynamical random environment leads to diffusive spreading and destroys linear and nonlinear localization. We find that maximal spreading is achieved for optimal phase fluctuation characteristics, which is a consequence of the competition between diffusion due to decoherence and ballistic transport within the mean free path distance.

Lyapunov Instabilities of Extended Systems

c

On the equilibrium state of random walkers in random environments: analytical results

We study equilibrium properties of random walkers in one-dimensional random environments of finite length L. From an exact expression for the quenched average of the free energy we derive analytical results for all cumulants and all Renyi entropies of the equilibrium distribution. In contrast to the finite variance of a typical non-equilibrium distribution in the unbiased situation, we find that in equilibrium the disorder averaged variance diverges with the size of the system as .

Disordered iterated maps: spectral properties, escape rates and anomalous transport

We investigate the transport properties of simple iterated maps with quenched disorder. The dynamics of these systems is mapped to random walks in random environments with next-nearest-neighbour transitions, constituting generalizations of the well-known Sinai model. The non-equilibrium properties are studied numerically by a direct observation of the transport behaviour, by investigating the density of states of the propagator and by considering the system-size dependence of the escape rate. Characteristic exponents associated with each of these quantities are determined and their dependence on the system parameters is evaluated. We find anomalously slow behaviour which in general deviates from the Sinai case and therefore generalizes the latter. These deviations are attributed to the generic absence of detailed balance, which implies that a potential can no longer be assigned.