Estimating covariant Lyapunov vectors from data.

Covariant Lyapunov vectors characterize the directions along which perturbations in dynamical systems grow. They have also been studied as predictors of critical transitions and extreme events. For many applications, it is necessary to estimate these vectors from data since model equations are unknown for many interesting phenomena. We propose an approach for estimating covariant Lyapunov vectors based on data records without knowing the underlying equations of the system. In contrast to previous approaches, our approach can be applied to high-dimensional datasets. We demonstrate that this purely data-driven approach can accurately estimate covariant Lyapunov vectors from data records generated by several low- and high-dimensional dynamical systems. The highest dimension of a time series from which covariant Lyapunov vectors are estimated in this contribution is 128.

Fluctuation-induced distributed resonances in oscillatory networks.

Across physics, biology, and engineering, the collective dynamics of oscillatory networks often evolve into self-organized operating states. How such networks respond to external fluctuating signals fundamentally underlies their function, yet is not well understood. Here, we present a theory of dynamic network response patterns and reveal how distributed resonance patterns emerge in oscillatory networks once the dynamics of the oscillatory units become more than one-dimensional. The network resonances are topology specific and emerge at an intermediate frequency content of the input signals, between global yet homogeneous responses at low frequencies and localized responses at high frequencies. Our analysis reveals why these patterns arise and where in the network they are most prominent. These results may thus provide general theoretical insights into how fluctuating signals induce response patterns in networked systems and simultaneously help to develop practical guiding principles for real-world network design and control.

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.

Vocal repertoire of long-finned pilot whales (Globicephala melas) in northern Norway.

The knowledge of the vocal repertoire of pilot whales is very limited. In this paper, the vocal repertoire of long-finned pilot whales recorded during different encounters in the Vestfjord in northern Norway between November 2006 and August 2010 are described. Sounds were analysed using two different methods: (1) an observer-based audio-visual inspection of FFT-derived spectrograms, with which, besides a general variety of clicks, buzzes, nonharmonic sounds, and whistles, 129 different distinct call types and 25 subtypes were distinguished. These call types included pulsed calls and discrete structured whistles varying from simple to highly complex structures composed of several segments and elements. In addition, ultrasonic whistles previously not described for pilot whales were found. In addition to the diversity of single calls, call sequences consisting of repetitions and combinations of specific call types were recorded and (2) a parametric approach that permitted the confirmation of the high variability in pilot whale call structures was developed. It is concluded that the pilot whale vocal repertoire is among the most complex for the mammalian species and the high structural variability, along with call repetitions and combinations, require a closer investigation to judge their importance for vocal communication.

Network susceptibilities: Theory and applications.

We introduce the concept of network susceptibilities quantifying the response of the collective dynamics of a network to small parameter changes. We distinguish two types of susceptibilities: vertex susceptibilities and edge susceptibilities, measuring the responses due to changes in the properties of units and their interactions, respectively. We derive explicit forms of network susceptibilities for oscillator networks close to steady states and offer example applications for Kuramoto-type phase-oscillator models, power grid models, and generic flow models. Focusing on the role of the network topology implies that these ideas can be easily generalized to other types of networks, in particular those characterizing flow, transport, or spreading phenomena. The concept of network susceptibilities is broadly applicable and may straightforwardly be transferred to all settings where networks responses of the collective dynamics to topological changes are essential.

Critical transitions and perturbation growth directions.

Critical transitions occur in a variety of dynamical systems. Here we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions. As suitable indicator variables for critical transitions, we consider changes in growth rates and directions of covariant Lyapunov vectors. Studying critical transitions in several models of fast-slow systems, i.e., a network of coupled FitzHugh-Nagumo oscillators, models for Josephson junctions, and the Hindmarsh-Rose model, we find that tangencies between covariant Lyapunov vectors are a common and maybe generic feature during critical transitions. We further demonstrate that this deviation from hyperbolic dynamics is linked to the occurrence of critical transitions by using it as an indicator variable and evaluating the prediction success through receiver operating characteristic curves. In the presence of noise, we find the alignment of covariant Lyapunov vectors and changes in finite-time Lyapunov exponents to be more successful in announcing critical transitions than common indicator variables as, e.g., finite-time estimates of the variance. Additionally, we propose a new method for estimating approximations of covariant Lyapunov vectors without knowledge of the future trajectory of the system. We find that these approximated covariant Lyapunov vectors can also be applied to predict critical transitions.

Critical Links and Nonlocal Rerouting in Complex Supply Networks.

Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet, it is still not well understood which links are particularly prone to inducing such outages. Here we analyze how the nature and location of each link impact the network's capability to maintain a stable supply. We propose two criteria to identify critical links on the basis of the topology and the load distribution of the network prior to link failure. They are determined via a link's redundant capacity and a renormalized linear response theory we derive. These criteria outperform the critical link prediction based on local measures such as loads. The results not only further our understanding of the physics of supply networks in general. As both criteria are available before any outage from the state of normal operation, they may also help real-time monitoring of grid operation, employing countermeasures and support network planning and design.

Quantifying group specificity of animal vocalizations without specific sender information.

Recordings of animal vocalization can lack information about sender and context. This is often the case in studies on marine mammals or in the increasing number of automated bioacoustics monitorings. Here, we develop a framework to estimate group specificity without specific sender information. We introduce and apply a bag-of-calls-and-coefficients approach (BOCCA) to study ensembles of cepstral coefficients calculated from vocalization signals recorded from a given animal group. Comparing distributions of such ensembles of coefficients by computing relative entropies reveals group specific differences. Applying the BOCCA to ensembles of calls recorded from group of long-finned pilot whales in northern Norway, we find that differences of vocalizations within social groups of pilot whales (Globicephala melas) are significantly lower than intergroup differences.

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 socioeconomic 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.

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 its 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 1996 model.

Predicting extreme avalanches in self-organized critical sandpiles.

In a finite-size Abelian sandpile model, extreme avalanches are repelling each other. Taking a time series of the avalanche size and using a decision variable derived from that, we predict the occurrence of a particularly large avalanche in the next time step. The larger the magnitude of these target avalanches, the better is their predictability. The predictability which is based on a finite-size effect, is discussed as a function of the system size.

Precursors of extreme increments.

We investigate precursors and the 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 autoregressive process of order 1, and numerically in wind speed recordings and long-range correlated autoregressive moving average processes 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.