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Several scenarios used in teaching feature a rolling motion with slipping that transitions to one without through friction with the ground. We summarise these transitions by introducing an unknown impulse that is transferred to the ground. Accounting for this in the conservation of angular and linear momentum, we can deduce the final state of these scenarios in a compact manner and without requiring details of the friction. Contrasting this technique with an explicit calculation of the friction illustrates how collisions and conservation laws allow to solve problems involving complex interactions by summarising them. We exemplify our technique with three scenarios: a moving ball starting to roll, a turning wheel being released on the ground, and a monowheel breaking.

We study numerically the dynamics of a network of all-to-all-coupled, identical sub-networks consisting of diffusively coupled, non-identical FitzHugh--Nagumo oscillators. For a large range of within- and between-network couplings, the network exhibits a variety of dynamical behaviors, previously described for single, uncoupled networks. We identify a region in parameter space in which the interplay of within- and between-network couplings allows for a richer dynamical behavior than can be observed for a single sub-network. Adjoining this atypical region, our network of networks exhibits transitions to multistability. We elucidate bifurcations governing the transitions between the various dynamics when crossing this region and discuss how varying the couplings affects the effective structure of our network of networks. Our findings indicate that reducing a network of networks to a single (but bigger) network might be not accurate enough to properly understand the complexity of its dynamics., Comment: 8 figures

We present a family of Python modules for the numerical integration of ordinary, delay, or stochastic differential equations. The key features are that the user enters the derivative symbolically and it is just-in-time-compiled, allowing the user to efficiently integrate differential equations from a higher-level interpreted language. The presented modules are particularly suited for large systems of differential equations such as used to describe dynamics on complex networks. Through the selected method of input, the presented modules also allow to almost completely automatize the process of estimating regular as well as transversal Lyapunov exponents for ordinary and delay differential equations. We conceptually discuss the modules' design, analyze their performance, and demonstrate their capabilities by application to timely problems.

We report on self-induced switchings between multiple distinct space--time patterns in the dynamics of a spatially extended excitable system. These switchings between low-amplitude oscillations, nonlinear waves, and extreme events strongly resemble a random process, although the system is deterministic. We show that a chaotic saddle -- which contains all the patterns as well as channel-like structures that mediate the transitions between them -- is the backbone of such a pattern switching dynamics. Our analyses indicate that essential ingredients for the observed phenomena are that the system behaves like an inhomogeneous oscillatory medium that is capable of self-generating spatially localized excitations and that is dominated by short-range connections but also features long-range connections. With our findings, we present an alternative to the well-known ways to obtain self-induced pattern switching, namely noise-induced attractor hopping, heteroclinic orbits, and adaptation to an external signal. This alternative way can be expected to improve our understanding of pattern switchings in spatially extended natural dynamical systems like the brain and the heart.

Extreme events occur in many spatially extended dynamical systems, often devastatingly affecting human life which makes their reliable prediction and efficient prevention highly desirable. We study the prediction and prevention of extreme events in a spatially extended system, a system of coupled FitzHugh-Nagumo units, in which extreme events occur in a spatially and temporally irregular way. Mimicking typical constraints faced in field studies, we assume not to know the governing equations of motion and to be able to observe only a subset of all phase-space variables for a limited period of time. Based on reconstructing the local dynamics from data and despite being challenged by the rareness of events, we are able to predict extreme events remarkably well. With small, rare, and spatiotemporally localized perturbations which are guided by our predictions, we are able to completely suppress extreme events in this system., Comment: 13 pages, 8 figures

We present a method that allows to distinguish between nearly periodic and strictly periodic time series. To this purpose, we employ a conservative criterion for periodicity, namely that the time series can be interpolated by a periodic function whose local extrema are also present in the time series. Our method is intended for the analysis of time series generated by deterministic time-continuous dynamical systems, where it can help telling periodic dynamics from chaotic or transient ones. We empirically investigate our method's performance and compare it to an approach based on marker events (or Poincar\'e sections). We demonstrate that our method is capable of detecting small deviations from periodicity and outperforms the marker-event-based approach in typical situations. Our method requires no adjustment of parameters to the individual time series, yields the period length with a precision that exceeds the sampling rate, and its runtime grows asymptotically linear with the length of the time series.

We explore the mathematical foundations of the vector space of physical dimensions introduced in A. Maksymowicz, Am. J. Phys. 44, 1976, and extend this formalism to the vector space of physical values. As different unit systems correspond to different bases of this vector space, our formalism may find use for introducing the concept of natural units and transforming physical values between unit systems., Comment: This is an author-created, un-copyedited version of an article published in the European Journal of Physics. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/0143-0807/36/3/035008

Systems of FitzHugh-Nagumo units with different coupling topologies are capable of self-generating and -terminating strong deviations from their regular dynamics that can be regarded as extreme events due to their rareness and recurrent occurrence. Here we demonstrate the crucial role of an interior crisis in the emergence of extreme events. In parameter space we identify this interior crisis as the organizing center of the dynamics by employing concepts of mixed-mode oscillations and of leaking chaotic systems. We find that extreme events occur in certain regions in parameter space, and we show the robustness of this phenomenon with respect to the system size.

Graph-theoretical analyses of complex brain networks is a rapidly evolving field with a strong impact for neuroscientific and related clinical research. Due to a number of confounding variables, however, a reliable and meaningful characterization of particularly functional brain networks is a major challenge. Addressing this problem, we present an analysis approach for weighted networks that makes use of surrogate networks with preserved edge weights or vertex strengths. We first investigate whether characteristics of weighted networks are influenced by trivial properties of the edge weights or vertex strengths (e.g., their standard deviations). If so, these influences are then effectively segregated with an appropriate surrogate normalization of the respective network characteristic. We demonstrate this approach by re-examining, in a time-resolved manner, weighted functional brain networks of epilepsy patients and control subjects derived from simultaneous EEG/MEG recordings during different behavioral states. We show that this surrogate-assisted analysis approach reveals complementary information about these networks, can aid with their interpretation, and thus can prevent deriving inappropriate conclusions.

We study deterministic systems, composed of excitable units of FitzHugh-Nagumo type, that are capable of self-generating and self-terminating strong deviations from their regular dynamics without the influence of noise or parameter change. These deviations are rare, short-lasting, and recurrent and can therefore be regarded as extreme events. Employing a range of methods we analyze dynamical properties of the systems, identifying features in the systems' dynamics that may qualify as precursors to extreme events. We investigate these features and elucidate mechanisms that may be responsible for the generation of the extreme events.

Network theory provides novel concepts that promise an improved characterization of interacting dynamical systems. Within this framework, evolving networks can be considered as being composed of nodes, representing systems, and of time-varying edges, representing interactions between these systems. This approach is highly attractive to further our understanding of the physiological and pathophysiological dynamics in human brain networks. Indeed, there is growing evidence that the epileptic process can be regarded as a large-scale network phenomenon. We here review methodologies for inferring networks from empirical time series and for a characterization of these evolving networks. We summarize recent findings derived from studies that investigate human epileptic brain networks evolving on timescales ranging from few seconds to weeks. We point to possible pitfalls and open issues, and discuss future perspectives., Comment: In press (Physica D)

We propose a Markov chain method to efficiently generate 'surrogate networks' that are random under the constraint of given vertex strengths. With these strength-preserving surrogates and with edge-weight-preserving surrogates we investigate the clustering coefficient and the average shortest path length of functional networks of the human brain as well as of the International Trade Networks. We demonstrate that surrogate networks can provide additional information about network-specific characteristics and thus help interpreting empirical weighted networks., Comment: 11 pages, 5 figures