Stochastic basins of attraction and generalized committor functions

Lindner, Michael; Hellmann, Frank

doi:10.1103/physreve.100.022124

Cited by 53 publications

(77 citation statements)

References 45 publications

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“…In case of very long transients, if the system has not truly converged, we may instead observe the basins of attraction of metastable states. See for example [14] for a related discussion of such basins. In this case the result will depend on simulation time.…”

Section: Integrationmentioning

confidence: 99%

Monte Carlo basin bifurcation analysis

2020

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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.

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Section: Integrationmentioning

confidence: 99%

Monte Carlo basin bifurcation analysis

2020

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“…In order to study the performance of various approaches, we make use of probabilistic methods [16][17][18][19][20][21][22][23][24] , that is, we define a scenario ensemble and evaluate the expected performance of the system with respect to the ensemble average by sampling over the ensemble. This approach allows us to study the properties of the system for the whole ensemble, rather than in individual case studies.…”

Section: Introductionmentioning

confidence: 99%

A multiplex, multi-timescale model approach for economic and frequency control in power grids

Strenge

Schultz

Kurths

et al. 2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Power systems are subject to fundamental changes due to the increasing infeed of decentralised renewable energy sources and storage. The decentralised nature of the new actors in the system requires new concepts for structuring the power grid, and achieving a wide range of control tasks ranging from seconds to days. Here we introduce a multiplex dynamical network model covering all control timescales. Crucially, we combine a decentralised, self-organised lowlevel control and a smart grid layer of devices that can aggregate information from remote sources. The safety-critical task of frequency control is performed by the former, the economic objective of demand matching dispatch by the latter. Having both aspects present in the same model allows us to study the interaction between the layers. Remarkably, we find that adding communication in the form of aggregation does not improve the performance in the cases considered. Instead, the self-organised state of the system already contains the information required to learn the demand structure in the entire grid. The model introduced here is highly flexible, and can accommodate a wide range of scenarios relevant to future power grids. We expect that it is especially useful in the context of low-energy microgrids with distributed generation.Highly decentralised power grids, possibly in the context of prosumer systems, require new concepts for their stable operation. We expect that both self-organised systems as well as intelligent devices with communication capability that can aggregate information from remote sources will play a central role. Here we introduce a multiplex network model that combines both aspects, and use it in a basic scenario and uncover surprising interactions between the layers.

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“…[31] Moreover, a QSH edge coupled with magnet/superconductor/magnet heterostructure is proposed to be able to probe the Majorana bound states inside the heterostructure. [32] Fractional QSH effect [33] is also predicted to host Majorana bound states, [34][35][36] providing another way to realize states for topological quantum computing.…”

Section: Introductionmentioning

confidence: 99%

Quantum Spin Hall Materials

Cao

Chen

2019

Adv Quantum Tech

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The quantum spin Hall (QSH) effect describes the state of matter in certain 2D electron systems, in which an insulating bulk state arises together with helical states at the edge of the sample. In stark contrast to its closest kin, the integer quantum Hall state, the QSH state exists only in time‐reversal symmetric system (e.g., in non‐magnetic materials and without the application of external magnetic field). This article reviews the development of the understanding and construction of the QSH states after their first theoretical proposal, with an emphasis on the materials perspective. Certain semiconductor quantum wells and 2D materials with strong spin–orbit coupling have been found to support QSH states.

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Stochastic basins of attraction and generalized committor functions

Cited by 53 publications

References 45 publications

Monte Carlo basin bifurcation analysis

Monte Carlo basin bifurcation analysis

A multiplex, multi-timescale model approach for economic and frequency control in power grids

Quantum Spin Hall Materials

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