Wolfram Just

Robustness of trans-European gas networks

Here, we uncover the load and fault-tolerant backbones of the trans-European gas pipeline network. Combining topological data with information on intercountry flows, we estimate the global load of the network and its tolerance to failures. To do this, we apply two complementary methods generalized from the betweenness centrality and the maximum flow. We find that the gas pipeline network has grown to satisfy a dual purpose. On one hand, the major pipelines are crossed by a large number of shortest paths thereby increasing the efficiency of the network; on the other hand, a nonoperational pipeline causes only a minimal impact on network capacity, implying that the network is error tolerant. These findings suggest that the trans-European gas pipeline network is robust, i.e., error tolerant to failures of high load links.

Limits of time-delayed feedback control

Abstract General features of stability domains for time-delayed feedback control exist, which can be predicted analytically. We clarify, why the control scheme with a single delay term can only stabilise orbits with short periods or small Lyapunov exponents, and derive a quantitative estimate. The limitation can be relaxed by employing multiple delay terms. In particular, the extended time delay autosynchronisation method is investigated in detail. Analytic calculations are in good agreement with results of numerical simulations and with experimental data from a nonlinear diode resonator.

Stochastic modelling: replacing fast degrees of freedom by noise.

In systems with timescale separation, where the fast degrees of freedom exhibit chaotic motion, the latter are replaced by suitable stochastic processes. A projection technique is employed to derive equations of motion for the phase space density of the slow variables by eliminating the fast ones. The resulting equations can be approximated in a controlled way by Fokker-Planck equations or equivalently by stochastic differential equations for slow degrees of freedom. We discuss some model situations and explore the accuracy of the approximations by numerical simulations.

Time-delay autosynchronization of the spatiotemporal dynamics in resonant tunneling diodes.

The double barrier resonant tunneling diode exhibits complex spatiotemporal patterns including low-dimensional chaos when operated in an active external circuit. We demonstrate how autosynchronization by time-delayed feedback control can be used to select and stabilize specific current density patterns in a noninvasive way. We compare the efficiency of different control schemes involving feedback in either local spatial or global degrees of freedom. The numerically obtained Floquet exponents are explained by analytical results from linear stability analysis.

Brownian motion with dry friction: Fokker–Planck approach

We solve a Langevin equation, first studied by de Gennes, in which there is a solid–solid or dry friction force acting on a Brownian particle in addition to the viscous friction usually considered in the study of Brownian motion. We obtain both the time-dependent propagator of this equation and the velocity correlation function by solving the associated time-dependent Fokker–Planck equation. Exact results are found for the case where only dry friction acts on the particle. For the case where both dry and viscous friction forces are present, series representations of the propagator and correlation function are obtained in terms of parabolic cylinder functions. Similar series representations are also obtained for the case where an external constant force is added to the Langevin equation.

Delayed complex systems: an overview

There does not exist a generally accepted definition for the notion of complex systems in science, but it is a common belief that complex systems show features that cannot be explained by just looking at their constituents. Thus, a complex system normally involves interaction of subunits, and

Self-stabilization of high frequency oscillations in semiconductor superlattices by time-delay autosynchronization

We present a scheme to stabilize high-frequency domain oscillations in semiconductor superlattices by a time-delayed feedback loop. Applying concepts from chaos control theory we propose to control the spatiotemporal dynamics of fronts of accumulation and depletion layers which are generated at the emitter and may collide and annihilate during their transit, and thereby suppress chaos. The proposed method only requires the feedback of internal global electrical variables, viz., current and voltage, which makes the practical implementation very easy.

Theoretical and experimental aspects of chaos control by time-delayed feedback

We review recent developments for the control of chaos by time-delayed feedback methods. While such methods are easily applied even in quite complex experimental context the theoretical analysis yields infinite-dimensional differential-difference systems which are hard to tackle. The essential ideas for a general theoretical approach are sketched and the results are compared to electronic circuits and to high power ferromagnetic resonance experiments. Our results show that the control performance can be understood on the basis of experimentally accessible quantities without resort to any model for the internal dynamics.

Elimination of Fast Chaotic Degrees of Freedom: On the Accuracy of the Born Approximation

We apply standard projection operator techniques known from nonequilibrium statistical mechanics to eliminate fast chaotic degrees of freedom in a low-dimensional dynamical system. Through the usual perturbative approach we end up in second order with a stochastic system where the fast chaotic degrees of freedom are modelled by Gaussian white noise. The accuracy of the perturbation expansion is analysed in detail by the discussion of an exactly solvable model.

Globally coupled maps: phase transitions and synchronization

Abstract Bifurcations in a system of coupled maps are investigated. Using symbolic dynamics it is shown that for coupled shift maps the well-known space-time-mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state. For coupled non-hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour. The anomalous dependence of fluctuations on the system size is attributed to these bifurcations.