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Author: Heitzig, J. × End date: 2019 × Start date: 2010 × Has files: Yes × Publisher: Bristol : Institute of Physics Publishing ×

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Now showing 1 - 3 of 3
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    Detours around basin stability in power networks
    (Bristol : Institute of Physics Publishing, 2014) Schultz, P.; Heitzig, J.; Kurths, J.
    To analyse the relationship between stability against large perturbations and topological properties of a power transmission grid, we employ a statistical analysis of a large ensemble of synthetic power grids, looking for significant statistical relationships between the single-node basin stability measure and classical as well as tailormade weighted network characteristics. This method enables us to predict poor values of single-node basin stability for a large extent of the nodes, offering a node-wise stability estimation at low computational cost. Further, we analyse the particular function of certain network motifs to promote or degrade the stability of the system. Here we uncover the impact of so-called detour motifs on the appearance of nodes with a poor stability score and discuss the implications for power grid design.
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    Timing of transients: Quantifying reaching times and transient behavior in complex systems
    (Bristol : Institute of Physics Publishing, 2017) Kittel, T.; Heitzig, J.; Webster, K.; Kurths, J.
    In dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which of them approaches the attractor earlier. In this article, we categorize several problems of quantifying such transient times. To treat them, we propose two metrics, area under distance curve and regularized reaching time, that capture two complementary aspects of transient dynamics. The first, area under distance curve, is the distance of the trajectory to the attractor integrated over time. It measures which trajectories are 'reluctant', i.e. stay distant from the attractor for long, or 'eager' to approach it right away. Regularized reaching time, on the other hand, quantifies the additional time (positive or negative) that a trajectory starting at a chosen initial condition needs to approach the attractor as compared to some reference trajectory. A positive or negative value means that it approaches the attractor by this much 'earlier' or 'later' than the reference, respectively. We demonstrated their substantial potential for application with multiple paradigmatic examples uncovering new features.
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    Potentials and limits to basin stability estimation
    (Bristol : Institute of Physics Publishing, 2017) Schultz, P.; Menck, P.J.; Heitzig, J.; Kurths, J.
    Stability assessment methods for dynamical systems have recently been complemented by basin stability and derived measures, i.e. probabilistic statements whether systems remain in a basin of attraction given a distribution of perturbations. Their application requires numerical estimation via Monte Carlo sampling and integration of differential equations. Here, we analyse the applicability of basin stability to systems with basin geometries that are challenging for this numerical method, having fractal basin boundaries and riddled or intermingled basins of attraction. We find that numerical basin stability estimation is still meaningful for fractal boundaries but reaches its limits for riddled basins with holes.