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Author: Heitzig, J. ×

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Now showing 1 - 10 of 11
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    Topology of sustainable management of dynamical systems with desirable states: From defining planetary boundaries to safe operating spaces in the Earth system
    (München : European Geopyhsical Union, 2016) Heitzig, J.; Kittel, T.; Donges, J.F.; Molkenthin, N.
    To keep the Earth system in a desirable region of its state space, such as defined by the recently suggested "tolerable environment and development window", "guardrails", "planetary boundaries", or "safe (and just) operating space for humanity", one needs to understand not only the quantitative internal dynamics of the system and the available options for influencing it (management) but also the structure of the system's state space with regard to certain qualitative differences. Important questions are, which state space regions can be reached from which others with or without leaving the desirable region, which regions are in a variety of senses "safe" to stay in when management options might break away, and which qualitative decision problems may occur as a consequence of this topological structure? In this article, we develop a mathematical theory of the qualitative topology of the state space of a dynamical system with management options and desirable states, as a complement to the existing literature on optimal control which is more focussed on quantitative optimization and is much applied in both the engineering and the integrated assessment literature. We suggest a certain terminology for the various resulting regions of the state space and perform a detailed formal classification of the possible states with respect to the possibility of avoiding or leaving the undesired region. Our results indicate that, before performing some form of quantitative optimization such as of indicators of human well-being for achieving certain sustainable development goals, a sustainable and resilient management of the Earth system may require decisions of a more discrete type that come in the form of several dilemmas, e.g. choosing between eventual safety and uninterrupted desirability, or between uninterrupted safety and larger flexibility. We illustrate the concepts and dilemmas drawing on conceptual models from climate science, ecology, coevolutionary Earth system modelling, economics, and classical mechanics, and discuss their potential relevance for the climate and sustainability debate, in particular suggesting several levels of planetary boundaries of qualitatively increasing safety.
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    Abrupt transitions in time series with uncertainties
    (London : Nature Publishing Group, 2018) Goswami, B.; Boers, N.; Rheinwalt, A.; Marwan, N.; Heitzig, J.; Breitenbach, S.F.M.; Kurths, J.
    Identifying abrupt transitions is a key question in various disciplines. Existing transition detection methods, however, do not rigorously account for time series uncertainties, often neglecting them altogether or assuming them to be independent and qualitatively similar. Here, we introduce a novel approach suited to handle uncertainties by representing the time series as a time-ordered sequence of probability density functions. We show how to detect abrupt transitions in such a sequence using the community structure of networks representing probabilities of recurrence. Using our approach, we detect transitions in global stock indices related to well-known periods of politico-economic volatility. We further uncover transitions in the El Niño-Southern Oscillation which coincide with periods of phase locking with the Pacific Decadal Oscillation. Finally, we provide for the first time an 'uncertainty-aware' framework which validates the hypothesis that ice-rafting events in the North Atlantic during the Holocene were synchronous with a weakened Asian summer monsoon.
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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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    Deciphering the imprint of topology on nonlinear dynamical network stability
    (Bristol : IOP Publishing, 2017) Nitzbon, J.; Schultz, P.; Heitzig, J.; Hellmann, F.
    Coupled oscillator networks show complex interrelations between topological characteristics of the network and the nonlinear stability of single nodes with respect to large but realistic perturbations. We extend previous results on these relations by incorporating sampling-based measures of the transient behaviour of the system, its survivability, as well as its asymptotic behaviour, its basin stability. By combining basin stability and survivability we uncover novel, previously unknown asymptotic states with solitary, desynchronized oscillators which are rotating with a frequency different from their natural one. They occur almost exclusively after perturbations at nodes with specific topological properties. More generally we confirm and significantly refine the results on the distinguished role tree-shaped appendices play for nonlinear stability. We find a topological classification scheme for nodes located in such appendices, that exactly separates them according to their stability properties, thus establishing a strong link between topology and dynamics. Hence, the results can be used for the identification of vulnerable nodes in power grids or other coupled oscillator networks. From this classification we can derive general design principles for resilient power grids. We find that striving for homogeneous network topologies facilitates a better performance in terms of nonlinear dynamical network stability. While the employed second-order Kuramoto-like model is parametrised to be representative for power grids, we expect these insights to transfer to other critical infrastructure systems or complex network dynamics appearing in various other fields.
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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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    Estimation of sedimentary proxy records together with associated uncertainty
    (Göttingen : Copernicus GmbH, 2015) Goswami, B.; Heitzig, J.; Rehfeld, K.; Marwan, N.; Anoop, A.; Prasad, S.; Kurths, J.
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    A matrix clustering method to explore patterns of land-cover transitions in satellite-derived maps of the Brazilian Amazon
    (Göttingen : Copernicus GmbH, 2017) Müller-Hansen, F.; Cardoso, M.F.; Dalla-Nora, E.L.; Donges, J.F.; Heitzig, J.; Kurths, J.; Thonicke, K.
    Changes in land-use systems in tropical regions, including deforestation, are a key challenge for global sustainability because of their huge impacts on green-house gas emissions, local climate and biodiversity. However, the dynamics of land-use and land-cover change in regions of frontier expansion such as the Brazilian Amazon are not yet well understood because of the complex interplay of ecological and socioeconomic drivers. In this paper, we combine Markov chain analysis and complex network methods to identify regimes of land-cover dynamics from land-cover maps (TerraClass) derived from high-resolution (30ĝ€m) satellite imagery. We estimate regional transition probabilities between different land-cover types and use clustering analysis and community detection algorithms on similarity networks to explore patterns of dominant land-cover transitions. We find that land-cover transition probabilities in the Brazilian Amazon are heterogeneous in space, and adjacent subregions tend to be assigned to the same clusters. When focusing on transitions from single land-cover types, we uncover patterns that reflect major regional differences in land-cover dynamics. Our method is able to summarize regional patterns and thus complements studies performed at the local scale.
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    Comparison of correlation analysis techniques for irregularly sampled time series
    (Göttingen : Copernicus GmbH, 2011) Rehfeld, K.; Marwan, N.; Heitzig, J.; Kurths, J.
    Geoscientific measurements often provide time series with irregular time sampling, requiring either data reconstruction (interpolation) or sophisticated methods to handle irregular sampling. We compare the linear interpolation technique and different approaches for analyzing the correlation functions and persistence of irregularly sampled time series, as Lomb-Scargle Fourier transformation and kernel-based methods. In a thorough benchmark test we investigate the performance of these techniques. All methods have comparable root mean square errors (RMSEs) for low skewness of the inter-observation time distribution. For high skewness, very irregular data, interpolation bias and RMSE increase strongly. We find a 40 % lower RMSE for the lag-1 autocorrelation function (ACF) for the Gaussian kernel method vs. the linear interpolation scheme,in the analysis of highly irregular time series. For the cross correlation function (CCF) the RMSE is then lower by 60 %. The application of the Lomb-Scargle technique gave results comparable to the kernel methods for the univariate, but poorer results in the bivariate case. Especially the high-frequency components of the signal, where classical methods show a strong bias in ACF and CCF magnitude, are preserved when using the kernel methods. We illustrate the performances of interpolation vs. Gaussian kernel method by applying both to paleo-data from four locations, reflecting late Holocene Asian monsoon variability as derived from speleothem δ18O measurements. Cross correlation results are similar for both methods, which we attribute to the long time scales of the common variability. The persistence time (memory) is strongly overestimated when using the standard, interpolation-based, approach. Hence, the Gaussian kernel is a reliable and more robust estimator with significant advantages compared to other techniques and suitable for large scale application to paleo-data.
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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.
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    A thought experiment on sustainable management of the earth system
    (Basel : MDPI AG, 2018) Heitzig, J.; Barfuss, W.; Donges, J.F.
    We introduce and analyze a simple formal thought experiment designed to reflect a qualitative decision dilemma humanity might currently face in view of anthropogenic climate change. In this exercise, each generation can choose between two options, either setting humanity on a pathway to certain high wellbeing after one generation of suffering, or leaving the next generation in the same state as the current one with the same options, but facing a continuous risk of permanent collapse. We analyze this abstract setup regarding the question of what the right choice would be both in a rationality-based framework including optimal control, welfare economics, and game theory, and by means of other approaches based on the notions of responsibility, safe operating spaces, and sustainability paradigms. Across these different approaches, we confirm the intuition that a focus on the long-term future makes the first option more attractive while a focus on equality across generations favors the second. Despite this, we generally find a large diversity and disagreement of assessments both between and within these different approaches, suggesting a strong dependence on the choice of the normative framework used. This implies that policy measures selected to achieve targets such as the United Nations Sustainable Development Goals can depend strongly on the normative framework applied and specific care needs to be taken with regard to the choice of such frameworks.