bSTAB: an open-source software for computing the basin stability of multi-stable dynamical systems

The pervasiveness of multi-stability in nonlinear dynamical systems calls for novel concepts of stability and a consistent quantification of long-term behavior. The basin stability is a global stability metric that builds on estimating the basin of attraction volumes by Monte Carlo sampling. The computation involves extensive numerical time integrations, attractor characterization, and clustering of trajectories. We introduce , an open-source software project that aims at enabling researchers to efficiently compute the basin stability of their dynamical systems with minimal efforts and in a highly automated manner. The source code, available at https://github.com/TUHH-DYN/bSTAB/, is available for the programming language featuring parallelization for distributed computing, automated sensitivity and bifurcation analysis as well as plotting functionalities. We illustrate the versatility and robustness of for four canonical dynamical systems from several fields of nonlinear dynamics featuring periodic and chaotic dynamics, complicated multi-stability, non-smooth dynamics, and fractal basins of attraction. The projects aims at fostering interdisciplinary scientific collaborations in the field of nonlinear dynamics and is driven by the interaction and contribution of the community to the software package.

Interactions of climate tipping elements and their associated basin stability in a conceptual model for tipping cascades

<p>With progressing global warming, there is an increased risk that one or several climate tipping elements might cross a critical threshold, resulting in severe consequences for the global climate, ecosystems and human societies. Here, we study a subset of four tipping elements and their interactions in a conceptual and easily extendable framework: the Greenland Ice Sheet, the West Antarctic Ice Sheet, the Atlantic Meridional Overturning Circulation (AMOC) and the Amazon rainforest.</p><p>In a large-scale Monte-Carlo simulation, we explicitly investigate the domino effects triggered by each of the individual tipping elements under global warming in equilibrium experiments. Thereby, we reveal the roles of each of the individual tipping elements in cascading transitions. Further, we perform a comprehensive basin stability analysis to detect the stable states of the interacting system and discuss their associated Earth system resilience. Finally, we analyse whether additional internal temperature feedbacks of the tipping elements might be able to increase the risk of triggering tipping events and cascades.</p><p>In our model experiments, we find: (i) the Greenland and the West Antarctic Ice Sheet are often the initiators of tipping cascades, while the AMOC typically takes on the role as a mediator of cascades. (ii) The interactions between the tipping elements considered here overall have a destabilizing effect on the climate system as a whole. (iii) In our model, the large ice sheets are of particular importance for the resilience of the Earth system on long time scales, as found by basin stability measures. (iv) Additional internal temperature feedbacks of the tipping elements can slightly increase the risk of triggering tipping events.</p>

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Stability considerations play a central role in structural dynamics to determine states that are robust against perturbations during the operation. Linear stability concepts, such as the complex eigenvalue analysis, constitute the core of analysis approaches in engineering reality. However, most stability concepts are limited to local perturbations, i.e., they can only measure a state’s stability against small perturbations. Recently, the concept of basin stability was proposed as a global stability concept for multi-stable systems. As multi-stability is a well-known property of a range of nonlinear dynamical systems, this work studies the basin stability of bi-stable mechanical oscillators that are affected and self-excited by dry friction. The results indicate how the basin stability complements the classical binary stability concepts for quantifying how stable a state is given a set of permissible perturbations.

The Basin Stability of Bi-Stable Friction-Excited Oscillators

Stability considerations play a central role in structural dynamics to determine states that are robust against perturbations during the operation. Linear stability concepts, such as the complex eigenvalue analysis, constitute the core of analysis approaches in engineering reality. However, most stability concepts are limited to local perturbations, i.e. they can only measure a state’s stability against small perturbations. Recently, the concept of basin stability has been proposed as a global stability concept for multi-stable systems. As multi-stability is a well-known property of a range of nonlinear dynamical systems, this work studies the basin stability of bi-stable mechanical oscillators that are affected and self-excited by dry friction. The results indicate how the basin stability complements the classical binary stability concepts for quantifying how stable a state is given a set of permissible perturbations.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

In this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.