Resilience of dynamical systems

Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining how "globally stable" a nonlinear system is very challenging. Over the last few decades, many different ideas have been developed to address this issue, primarily driven by concrete applications. In particular, several disciplines suggested a web of concepts under the headline "resilience". Unfortunately, there are many different variants and explanations of resilience, and often the definitions are left relatively vague, sometimes even deliberately. Yet, to allow for a structural development of a mathematical theory of resilience that can be used across different areas, one has to ensure precise starting definitions and provide a mathematical comparison of different resilience measures. In this work, we provide a systematic review of the most relevant indicators of resilience in the context of continuous dynamical systems, grouped according to their mathematical features. The indicators are also generalized to be applicable to any attractor. These steps are important to ensure a more reliable, quantitatively comparable and reproducible study of resilience in dynamical systems. Furthermore, we also develop a new concept of resilience against certain non-autonomous perturbations to demonstrate, how one can naturally extend our framework. All the indicators are finally compared via the analysis of a classic scalar model from population dynamics to show that direct quantitative application-based comparisons are an immediate consequence of a detailed mathematical analysis.Comment: 54 pages, 18 figure

Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

If data are generated by a system with a d-dimensional attractor, then Takens’ theorem guarantees that reconstruction that is diffeomorphic to the original attractor can be built from the single time series in 2d+1-dimensional phase space. However, under certain conditions, reconstruction is possible even in a space of smaller dimension. This topic is very important because the size of the reconstruction space relates to the effectiveness of the whole subsequent analysis. In this paper, the false nearest neighbour (FNN) methods are revisited to estimate the optimum embedding parameters and the most appropriate observables for state space reconstruction. A modification of the false nearest neighbour method is introduced. The findings contribute to evidence that the length of the embedding time window (TW) is more important than the reconstruction delay time and the embedding dimension (ED) separately. Moreover, if several time series of the same system are observed, the choice of the one that is used for the reconstruction could also be critical. The results are demonstrated on two chaotic benchmark systems

Impact of chromium, arsenic and selected environmental variables on the vegetation and soil seed bank of subsidence basins

As a result of hard coal mining, specific areas of subsidence basins, which are often waterlogged, were created in the Czech part of the Upper Silesian Coal District. These areas may not have a high ecological value or, on the contrary, promising biotopes may arise here under suitable conditions, which should be our goal. As part of this work, the above-ground vegetation, and the soil seed bank (using the cultivation method) were investigated. The concentration of the risk elements chromium and arsenic, which are usually increased in mining areas, was determined by neutron activation analysis. Other environmental variables that can affect both the vegetation and the soil seed bank of subsidence basins (fine earth, slope, biotopes, and their representation in the vicinity) were also determined. Using multivariate DCA analysis, a statistically significant influence of chromium concentration and other variables on above-ground vegetation, characterized by the occurrence of many metallophytes and their high coverage, was found. The soil seed bank, which can impact the further development of vegetation, is mainly influenced by forest and wetland biotopes and the representation of areas covered by tailings in the vicinity of sampling sites. It is evident that there are species capable of resisting or accumulating chromium and arsenic pollution in both the above-ground vegetation and the soil seed bank, which can lead to the gradual rehabilitation of subsidence basins. This research can lead to a better understanding of the development of subsidence basins to increase their future ecological values.Web of Science17241240