Filters

2016 - 20173

More filters

2019 - 20203
Reset filters

Settings

Author: Heitzig, J. × Subject: Dynamical systems ×

Search Results

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    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.