Impact of symmetry on ergodic properties of triangular billiards.

Polygonal billiards constitute some of the simplest yet counterintuitive dynamical systems in physics. Even basic features of the dynamics, such as ergodicity of the microcanonical distribution or the decay of correlations have not been settled in general. In this Letter, we will highlight the importance of symmetries of the billiard table for the resulting dynamics. Although typical triangular billiards appear to show correlation decay, symmetric billiards may not even be ergodic with respect to the uniform distribution in phase space.

Nonequilibrium dynamics of a pure dry friction model subjected to colored noise.

We investigate the impact of noise on a two-dimensional simple paradigmatic piecewise-smooth dynamical system. For that purpose, we consider the motion of a particle subjected to dry friction and colored noise. The finite correlation time of the noise provides an additional dimension in phase space, causes a nontrivial probability current, and establishes a proper nonequilibrium regime. Furthermore, the setup allows for the study of stick-slip phenomena, which show up as a singular component in the stationary probability density. Analytic insight can be provided by application of the unified colored noise approximation, developed by Jung and Hänggi [Phys. Rev. A 35, 4464(R) (1987)0556-279110.1103/PhysRevA.35.4464]. The analysis of probability currents and of power spectral densities underpins the observed stick-slip transition, which is related with a critical value of the noise correlation time.

Synchronization of Heterogeneous Oscillators by Noninvasive Time-Delayed Cross Coupling.

We demonstrate that nonidentical systems, in particular, nonlinear oscillators with different time scales, can be synchronized if a mutual coupling via time-delayed control signals is implemented. Each oscillator settles on an unstable state, say a fixed point or an unstable periodic orbit, with a coupling force which vanishes in the long time limit. We present the underlying theoretical considerations and numerical simulations, and, moreover, demonstrate the concept experimentally in nonlinear electronic oscillators.

Large-deviation properties of Brownian motion with dry friction.

We investigate piecewise-linear stochastic models with regard to the probability distribution of functionals of the stochastic processes, a question that occurs frequently in large deviation theory. The functionals that we are looking into in detail are related to the time a stochastic process spends at a phase space point or in a phase space region, as well as to the motion with inertia. For a Langevin equation with discontinuous drift, we extend the so-called backward Fokker-Planck technique for non-negative support functionals to arbitrary support functionals, to derive explicit expressions for the moments of the functional. Explicit solutions for the moments and for the distribution of the so-called local time, the occupation time, and the displacement are derived for the Brownian motion with dry friction, including quantitative measures to characterize deviation from Gaussian behavior in the asymptotic long time limit.

First-passage time of Brownian motion with dry friction.

We provide an analytic solution to the first-passage time (FPT) problem of a piecewise-smooth stochastic model, namely Brownian motion with dry friction, using two different but closely related approaches which are based on eigenfunction decompositions on the one hand and on the backward Kolmogorov equation on the other. For the simple case containing only dry friction, a phase-transition phenomenon in the spectrum is found which relates to the position of the exit point, and which affects the tail of the FPT distribution. For the model containing as well a driving force and viscous friction the impact of the corresponding stick-slip transition and of the transition to ballistic exit is evaluated quantitatively. The proposed model is one of the very few cases where FPT properties are accessible by analytical means.

Weak-noise limit of a piecewise-smooth stochastic differential equation.

We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a simple model of Brownian motion with solid friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided the singularity of the path integral associated with the nonsmooth SDE is treated with some heuristics. We also show that, as in the case of smooth SDEs, the deterministic paths of the noiseless system correctly describe the behavior of the nonsmooth SDE in the low-noise limit. Finally, we consider a smooth regularization of the piecewise-constant SDE and study to what extent this regularization can rectify some of the problems encountered when dealing with discontinuous drifts and singularities in SDEs.

Fair sharing of resources in a supply network with constraints.

This paper investigates the effect of network topology on the fair allocation of network resources among a set of agents, an all-important issue for the efficiency of transportation networks all around us. We analyze a generic mechanism that distributes network capacity fairly among existing flow demands. The problem can be solved by semianalytical methods on a nearest-neighbor graph with one source and sink pair, when transport occurs over shortest paths. For this setup, we uncover a broad range of patterns of intersecting shortest paths as a function of the distance between the source and the sink. When the number of intersections is the maximum and the distance between the source and the sink is large, we find that a fair allocation implies a decrease of at least 50% from the maximum throughput. We also find that the histogram of the flow allocations assigned to the agents decays as a power law with exponent -1. Our semianalytical framework suggests possible explanations for the well-known reduction of the throughput in fair allocations. It also suggests that the combination of network topology and routing rules can lead to highly uneven (but fair) distributions of resources, a remark of caution to network designers.

Coexisting patterns of population oscillations: the degenerate Neimark-Sacker bifurcation as a generic mechanism.

We investigate a population dynamics model that exhibits a Neimark-Sacker bifurcation with a period that is naturally close to 4. Beyond the bifurcation, the period soon becomes locked at 4 due to a strong resonance, and a second attractor of period 2 emerges, which coexists with the first attractor over a considerable parameter range. A linear stability analysis and a numerical investigation of the second attractor reveal that the bifurcations producing the second attractor occur naturally in this type of system.

A three-species model explaining cyclic dominance of Pacific salmon.

The four-year oscillations of the number of spawning sockeye salmon (Oncorhynchus nerka) that return to their native stream within the Fraser River basin in Canada are a striking example of population oscillations. The period of the oscillation corresponds to the dominant generation time of these fish. Various-not fully convincing-explanations for these oscillations have been proposed, including stochastic influences, depensatory fishing, or genetic effects. Here, we show that the oscillations can be explained as an attractor of the population dynamics, resulting from a strong resonance near a Neimark Sacker bifurcation. This explains not only the long-term persistence of these oscillations, but also reproduces correctly the empirical sequence of salmon abundance within one period of the oscillations. Furthermore, it explains the observation that these oscillations occur only in sockeye stocks originating from large oligotrophic lakes, and that they are usually not observed in salmon species that have a longer generation time.

Dynamic structure factors and Lyapunov modes in disordered chains.

We investigate various dynamic structure factors for harmonic and anharmonic chains. For harmonic chains with mass disorder we find some unexpected features, such as a fine structure contributing to a central peak, which is present also in the spatial spectra of the eigenfunctions. These results are contrasted with structure factors of Lyapunov modes obtained for the disordered Lennard-Jones chain. For this nonlinear system the static and the dynamic Lyapunov structure factors show opposite trends in their temperature dependence.

Experimental verification of Pyragas-Schöll-Fiedler control.

We present an experimental realization of time-delayed feedback control proposed by Schöll and Fiedler. The scheme enables us to stabilize torsion-free periodic orbits in autonomous systems, and to overcome the so-called odd number limitation. The experimental control performance is in quantitative agreement with the bifurcation analysis of simple model systems. The results uncover some general features of the control scheme which are deemed to be relevant for a large class of setups.

Stochastic modelling of intermittency.

Recently, methods have been developed to model low-dimensional chaotic systems in terms of stochastic differential equations. We tested such methods in an electronic circuit experiment. We aimed to obtain reliable drift and diffusion coefficients even without a pronounced time-scale separation of the chaotic dynamics. By comparing the analytical solutions of the corresponding Fokker-Planck equation with experimental data, we show here that crisis-induced intermittency can be described in terms of a stochastic model which is dominated by state-space-dependent diffusion. Further on, we demonstrate and discuss some limits of these modelling approaches using numerical simulations. This enables us to state a criterion that can be used to decide whether a stochastic model will capture the essential features of a given time series.

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.

Noise-free stochastic resonance at an interior crisis.

We report on the observation of noise-free stochastic resonance in an externally driven diode resonator close to an interior crisis. At sufficiently high excitation amplitudes the diode resonator shows a strange attractor which after the collision with an unstable period-three orbit exhibits crisis-induced intermittency. In the intermittency regime the system jumps between the previously stable chaotic attractor and the phase space region which has been made accessible by the crisis. This random process can be used to amplify a weak periodic signal through the mechanism of stochastic resonance. In contrast to conventional stochastic resonance no external noise is needed. The chaotic intrinsic dynamics plays the role of the stochastic forcing. Our data obtained from the diode resonator are compared with numerical simulations of the logistic map where a similar crisis-induced intermittency is observed.

Global properties in an experimental realization of time-delayed feedback control with an unstable control loop.

We demonstrate by electronic circuit experiments the feasibility of an unstable control loop to stabilize torsion-free orbits by time-delayed feedback control. Corresponding analytical normal form calculations and numerical simulations reveal a severe dependence of the basin of attraction on the particular coupling scheme of the control force. Such theoretical predictions are confirmed by the experiments and emphasize the importance of the coupling scheme for the global control performance.

Stochastic modeling of experimental chaotic time series.

Methods developed recently to obtain stochastic models of low-dimensional chaotic systems are tested in electronic circuit experiments. We demonstrate that reliable drift and diffusion coefficients can be obtained even when no excessive time scale separation occurs. Crisis induced intermittent motion can be described in terms of a stochastic model showing tunneling which is dominated by state space dependent diffusion. Analytical solutions of the corresponding Fokker-Planck equation are in excellent agreement with experimental data.

Fast Hamiltonian chaos: Heat bath without thermodynamic limit.

Chaotic Hamiltonian systems with time scale separation display features known from nonequilibrium statistical physics even when no thermodynamic limit is involved. In particular, fast chaotic degrees of freedom can be modeled by suitable stochastic forces and a Fokker-Planck equation governing the slow parts of the motion can be derived. It turns out that the underlying Hamiltonian structure results in fluctuation-dissipation relations which link the parameters of the effective stochastic model. Such properties are crucial to ensure the correct stationary state of the stochastic description. Our results demonstrate that concepts from thermodynamics can be transferred to dynamical systems with few degrees of freedom.

Phase transitions in coupled map lattices and in associated probabilistic cellular automata.

Analytical tools are applied to investigate piecewise linear coupled map lattices in terms of probabilistic cellular automata. The so-called disorder condition of probabilistic cellular automata is closely related with attracting sets in coupled map lattices. The importance of this condition for the suppression of phase transitions is illustrated by spatially one-dimensional systems. Invariant densities and temporal correlations are calculated explicitly. Ising type phase transitions are found for one-dimensional coupled map lattices acting on repelling sets and for a spatially two-dimensional Miller-Huse-like system with stable long time dynamics. Critical exponents are calculated within a finite size scaling approach. The relevance of detailed balance of the resulting probabilistic cellular automaton for the critical behavior is pointed out.

Accuracy and efficiency of reduced stochastic models for chaotic Hamiltonian systems with time-scale separation.

The study of the long time behavior of systems with time-scale separation is considerably facilitated if the fast degrees of freedom can be eliminated without affecting the slow dynamics. We investigate a technique in which the fluctuations due to a fast chaotic subsystem are replaced by a suitable stochastic process so that a Fokker-Planck equation for the slow variables results. The accuracy and efficiency of this technique is verified by the detailed numerical investigation of several coupled systems. The asymptotic behavior as well as transients turn out to be well modeled by the reduced dynamics. We concentrate on low-dimensional problems and cover different types of coupling schemes as well as different chaotic subsystems. As a physical application we discuss the classical dynamics of a hydrogen atom in a strong magnetic field.