Predictability of Critical Transitions

Critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socio-economic changes and climate transitions between ice-ages and warm-ages. From bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations. The consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition. However especially in the presence of noise it is not clear, whether these changes in observation variables are statistically relevant such that they could be used as indicators for critical transitions. In this contribution we investigate the predictability of critical transitions in conceptual models. We study the quadratic integrate-and-fire model and the van der Pol model, under the influence of external noise. We focus especially on the statistical analysis of the success of predictions and the overall predictability of the system. The performance of different indicator variables turns out to be dependent on the specific model under study and the conditions of accessing it. Furthermore, we study the influence of the magnitude of transitions on the predictive performance

Fluctuation-induced Distributed Resonances in Oscillatory Networks

Self-organized network dynamics prevails for systems across physics, biology and engineering. How external signals generate distributed responses in networked systems fundamentally underlies their function, yet is far from fully understood. Here we analyze the dynamic response patterns of oscillatory networks to fluctuating input signals. We disentangle the impact of the signal distribution across the network, the signals' frequency contents and the network topology. We analytically derive qualitatively different dynamic response patterns and find three frequency regimes: homogeneous responses at low frequencies, topology-dependent resonances at intermediate frequencies, and localized responses at high frequencies. The theory faithfully predicts the network-wide collective responses to regular and irregular, localized and distributed simulated signals, as well as to real input signals to power grids recorded from renewable-energy supplies. These results not only provide general insights into the formation of dynamic response patterns in networked systems but also suggest regime- and topology-specific design principles underlying network function.Comment: 7 pages, 4 figure

High-Intensity Discharge Lamp and Duffing Oscillator - Similarities and Differences

The processes inside the arc tube of high-intensity discharge lamps are investigated by finite element simulations. The behavior of the gas mixture inside the arc tube is governed by differential equations describing mass, energy and charge conservation as well as the Helmholtz equation for the acoustic pressure and the Navier-Stokes equation for the flow driven by the buoyancy and the acoustic streaming force. The model is highly nonlinear and requires a recursion procedure to account for the impact of acoustic streaming on the temperature and other fields. The investigations reveal the presence of a hysteresis and the corresponding jump phenomenon, quite similar to a Duffing oscillator. The similarities and, in particular, the differences of the nonlinear behavior of the high-intensity discharge lamp to that of a Duffing oscillator are discussed. For large amplitudes the high-intensity discharge lamp exhibits a stiffening effect in contrast to the Duffing oscillator.Comment: 14 pages, 8 figure

Logarithmic bred vectors in spatiotemporal chaos: structure and growth

Bred vectors are a type of finite perturbation used in prediction studies of atmospheric models that exhibit spatially extended chaos. We study the structure, spatial correlations, and the growth- rates of logarithmic bred vectors (which are constructed by using a given norm). We find that, after a suitable transformation, logarithmic bred vectors are roughly piecewise copies of the leading Lyapunov vector. This fact allows us to deduce a scaling law for the bred vector growth rate as a function of their amplitude. In addition, we relate growth rates with the spectrum of Lyapunov exponents corresponding to the most expanding directions. We illustrate our results with simulations of the Lorenz '96 model.Comment: 8 pages, 8 figure

Predictability of Extreme Events in Time Series

In this thesis we access the prediction of extreme events observing precursory structures, which were identified using a maximum likelihood approach. The main goal of this thesis is to investigate the dependence of the quality of a prediction on the magnitude of the events under study. Until now, this dependence was only sporadically reported for different phenomena without being understood as a general feature of predictions. We propose the magnitude dependence as a property of a prediction, indicating, whether larger events can be better, harder or equally well predicted than smaller events. Furthermore we specify a condition which can characterize the magnitude dependence of a distinguished measure for the quality of a prediction, the Receiver Operator characteristic curve (ROC). This test condition allows to relate the magnitude dependence of ap rediction task to the joint PDF of events and precursory variables. If we are able to describe the numerical estimate of this joint PDF by an analytic expression, we can not only characterize the magnitude dependence of events observed so far, but infer the magnitude dependence of events, larger then the observed events. Having the test condition specified, we study the magnitude dependence for the prediction of increments and threshold crossings in sequences of random variables and short- and long-range correlated stochastic processes. In dependence of the distribution of the process under study we obtain different magnitude dependences for the prediction of increments in Gaussian, exponentially, symmetrized exponentially, power-law and symmetrized power-law distributed processes. For threshold crossings we obtain the same magnitude dependence for all distributions studied. Furthermore we study the dependence on the event magnitude for the prediction of increments and threshold crossings in velocity increments, measured in a free jet flow and in wind-speed measurements. Additionally we introduce a method of post-processing the output of ensemble weather forecast models in order to identify precursory behavior, which could indicate failures of weather forecasts. We then study not only the success of this method, but also the magnitude dependence. keywords: extreme events, statistical inference, prediction via precursors, ROC curves, likelihood ratio, magnitude dependenc

Model-free inference of direct network interactions from nonlinear collective dynamics

The topology of interactions in network dynamical systems fundamentally underlies their function. Accelerating technological progress creates massively available data about collective nonlinear dynamics in physical, biological, and technological systems. Detecting direct interaction patterns from those dynamics still constitutes a major open problem. In particular, current nonlinear dynamics approaches mostly require to know a priori a model of the (often high dimensional) system dynamics. Here we develop a model-independent framework for inferring direct interactions solely from recording the nonlinear collective dynamics generated. Introducing an explicit dependency matrix in combination with a block-orthogonal regression algorithm, the approach works reliably across many dynamical regimes, including transient dynamics toward steady states, periodic and non-periodic dynamics, and chaos. Together with its capabilities to reveal network (two point) as well as hypernetwork (e.g., three point) interactions, this framework may thus open up nonlinear dynamics options of inferring direct interaction patterns across systems where no model is known.Comment: 10 pages, 7 figure

Reactions to extreme events: moving threshold model

In spite of precautions to avoid the harmful effects of extreme events, we experience recurrently phenomena that overcome the preventive barriers. These barriers usually increase drastically right after the occurrence of such extreme events, but steadily decay in their absence. In this paper we consider a simple model that mimics the evolution of the protection barriers to study the efficiency of the system's reaction to extreme events and how it changes our perception of the sequence of extreme events itself. We obtain that the usual method of fighting extreme events introduces a periodicity in their occurrence and is generally less efficient than the use of a constant barrier. On the other hand, it shows a good adaptation to the presence of slow non-stationarities.Comment: 14 pages and 7 figure

Precursors of extreme increments

We investigate precursors and predictability of extreme increments in a time series. The events we are focusing on consist in large increments within successive time steps. We are especially interested in understanding how the quality of the predictions depends on the strategy to choose precursors, on the size of the event and on the correlation strength. We study the prediction of extreme increments analytically in an AR(1) process, and numerically in wind speed recordings and long-range correlated ARMA data. We evaluate the success of predictions via receiver operator characteristics (ROC-curves). Furthermore, we observe an increase of the quality of predictions with increasing event size and with decreasing correlation in all examples. Both effects can be understood by using the likelihood ratio as a summary index for smooth ROC-curves