Monte Carlo basin bifurcation analysis

Abstract

Many high-dimensional complex systems exhibit an enormously complex landscape of possible asymptotic states. Here, we present a numerical approach geared towards analyzing such systems. It is situated between the classical analysis with macroscopic order parameters and a more thorough, detailed bifurcation analysis. With our machine learning method, based on random sampling and clustering methods, we are able to characterize the different asymptotic states or classes thereof and even their basins of attraction. In order to do this, suitable, easy to compute, statistics of trajectories with randomly generated initial conditions and parameters are clustered by an algorithm such as DBSCAN. Due to its modular and flexible nature, our method has a wide range of possible applications in many disciplines. While typical applications are oscillator networks, it is not limited only to ordinary differential equation systems, every complex system yielding trajectories, such as maps or agent-based models, can be analyzed, as we show by applying it the Dodds-Watts model, a generalized SIRS-model, modeling social and biological contagion. A second order Kuramoto model, used, e.g. to investigate power grid dynamics, and a Stuart-Landau oscillator network, each exhibiting a complex multistable regime, are shown as well. The method is available to use as a package for the Julia language.world applications: the Dodds-Watts model of social and biological contagion, a network of second order Kuramoto oscillators, used e.g. to model power grids and a network of Stuart-Landau oscillators, of importance for many chemical and biological systems, will follow in section 3. Lastly, these results and the performance and applicability of the presented method will be discussed in section 4.

On the approximation of basins of attraction using deep neural networks

“…In such cases, it is possible that more complex network architectures will be needed. If the system under consideration is high dimensional and the number of the coexisting attractors is not known a priori, the additional step of determining them will be necessary, using methods such as Monte Carlo basin bifurcation analysis 39 .…”

Ambient Forcing: Sampling Local Perturbations in Constrained Phase Spaces

“…A more detailed load model is needed to properly understand this phenomena but a fast energy storage at the loads could be used to change the power flows in time and mitigate the infeasabilities. In a future investigation Monte Carlo basin bifurcation analysis [26] could be used to investigate the different asymptotic states in such systems and their basin of attraction.…”

Analysis of a bistable climate toy model with physics-based machine learning methods

“…When analysing and working with high-dimensional models, knowledge of the largest basins of attractions is instrumental to understanding the model itself. The recently introduced Monte Carlo Basin Bifurcation Analysis (MCBB) [27] is a numerical method tailored for analysing basins of attraction of high-dimensional systems and how they vary when control parameters of the system are changed. The aim of MCBB is to find classes of similar attractors of a high-dimensional system that collectively have the largest basin of attraction with respect to a measure of initial conditions ρ 0 and how these classes of attractors and their basin volumes change when a control parameter p is changed in a range I p .…”

“…For high-dimensional systems, the integration far outweighs the computation of the distance matrix. For more details of MCBB, the reader is referred to [27].…”

“…We proceed as follows. The Monte Carlo Basin Bifurcation Analysis (MCBB) [27] is used to identify the basins of attractions with the largest volumes and how the volume changes when control parameters of the system are changed. MCBB makes use of random sampling and clustering techniques to quantify the volume of the basins of attraction and track how they change when control parameters of the system are varies.…”