On the application of set‐oriented numerical methods in the analysis of railway vehicle dynamics

Neumann, Nicolai; Goldschmidt, Stefan; Wallaschek, Jörg

doi:10.1002/pamm.200410270

Cited by 2 publications

(2 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This question leads to Eq. (19). On the other hand the transient behaviours might be important as well, that is in which way does the system return to the equilibrium states, how long does it take to do so and does it spend a long time in certain metastable states before reaching equilibrium.…”

Section: A Sojourn Timesmentioning

confidence: 99%

See 1 more Smart Citation

Stochastic basins of attraction and generalized committor functions

Lindner

Hellmann

2019

Phys. Rev. E

View full text Add to dashboard Cite

We generalize the concept of basin of attraction of a stable state in order to facilitate the analysis of dynamical systems with noise and to assess stability properties of metastable states and long transients. To this end we examine the notions of mean sojourn times and absorption probabilities for Markov chains and study their relation to the basins of attraction. Our approach is directly applicable to all systems that can be approximated as Markov chains, including stochastic and deterministic differential equations. We also provide a sampling based generalization of basin stability that works without resorting to the Markov approximation by sampling trajectories directly.We discuss two far reaching generalizations of the basin of attraction of an attractor. These apply to general sets in the phase space of nondeterministic systems. The first is based on absorption probabilities, the second on expected mean sojourn times with respect to a finite time horizon. By casting the problem in the transfer operator language, we are able to give a simple formula for the first generalization along the lines of committor functions.We show that the two notions of generalized basin coincide in the limit of a vanishing absorption probability and an infinite time horizon respectively. Importantly, for well-behaved deterministic systems this limit recovers the usual notion of basin of attraction. Finally we point out that derived quantities like the volume of the generalized basin are accessible through sampling trajectories at the same computational cost as evaluating basin stability for deterministic systems.

show abstract

Section: A Sojourn Timesmentioning

confidence: 99%

“…While these quantities can be defined for systems on a continuous state space as well, they originate in the theory of finite Markov chains, where their analysis is well understood. Commitor functions have been used previously as a tool to study deterministic basins, and to optimize basin stability in [16][17][18][19] .…”

Section: Introductionmentioning

confidence: 99%