Bounding the first exit from the basin: Independence times and finite-time basin stability

Schultz, Pamela N.; Hellmann, Frank; Webster, Kevin; Kurths, Jürgen

doi:10.1063/1.5013127

Cited by 21 publications

(17 citation statements)

References 40 publications

(33 reference statements)

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“…Compared to the work of Serdukova et al 12 our definition is entirely intrinsic and does not presupose knowing the basin of attraction. This allows for a straightforward estimator for the generalized basin stability, whereas it is not known whether such an estimator exists for the definition of Serdukova et al 12 (see Schultz et al 45 though for an estimator for a related quantity). We define stochastic basins more generally for measureable sets in arbitrary stochastic systems given that their evolution is described by a Markov operator.…”

Section: Discussionmentioning

confidence: 99%

Stochastic basins of attraction and generalized committor functions

Lindner

Hellmann

2019

Phys. Rev. E

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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.

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Section: Discussionmentioning

confidence: 99%

Stochastic basins of attraction and generalized committor functions

Lindner

Hellmann

2019

Phys. Rev. E

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“…We count distances below and above a certain threshold ε as successes and failures respectively, interpreting it as a Bernoulli trial to provide a rigorous confidence interval. This leads to a stability curve similar to those underlying the approach in [20].…”

Section: Discussionmentioning

confidence: 66%

“…To obtain a rigorous statement about the systems performance, we use the second distance measure (10). Rather than fixing a single , we can plot dρ, and its 95% confidence interval, as a function of , similar to [20]. This is shown in Fig.…”

Section: Tuning Pipeline Step Dρmentioning

confidence: 99%

Probabilistic Behavioral Distance and Tuning - Reducing and aggregating complex systems

Hellmann¹,

Ekaterina²,

Kurths³

et al. 2021

Preprint

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Given a complex system with a given interface to the rest of the world, what does it mean for a the system to behave close to a simpler specification describing the behavior at the interface? We give several definitions for useful notions of distances between a complex system and a specification by combining a behavioral and probabilistic perspective. These distances can be used to tune a complex system to a specification. We show that our approach can successfully tune non-linear networked systems to behave like much smaller networks, allowing us to aggregate large sub-networks into one or two effective nodes. Finally, we discuss similarities and differences between our approach and H∞ model reduction.

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“…Furthermore, as we have shown with the second example, the MiFaS approach works also well if not all state variables but only a subspace of perturbations is accessible-a problem relevant to many technical and natural systems. This paves the way to extent this method to single-node and multiple-node versions which enable us to compare the results with corresponding variations of other global stability techniques like basin stability 32,33,64 .…”

Section: Discussionmentioning

confidence: 99%

Minimal fatal shocks in multistable complex networks

Halekotte

Feudel

2020

Sci Rep

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Multistability is a common phenomenon which naturally occurs in complex networks. often one of the coexisting stable states can be identified as being the desired one for a particular application. We present here a global approach to identify the minimal perturbation which will instantaneously kick the system out of the basin of attraction of its desired state and hence induce a critical or fatal transition we call shock-tipping. the corresponding Minimal Fatal Shock is a vector whose length can be used as a global stability measure and whose direction in state space allows us to draw conclusions on weaknesses of the network corresponding to critical network motifs. We demonstrate this approach in plant-pollinator networks and the power grid of Great Britain. in both system classes, tree-like substructures appear to be the most vulnerable with respect to the minimal shock perturbation. Many processes in nature can be well described in terms of nonlinear dynamical systems possessing multiple stable states for constant external conditions 1,2. As long as perturbations are small, it is sufficient to consider the linearized problem around an attracting state to evaluate its 'local' stability. However, in real systems, the most threatening perturbations are usually not small but large, even constituting extreme events like storms, earthquakes or financial crises 3-5. Obviously, large perturbations in multistable systems call for a 'global' stability paradigm. Adequate frameworks have been introduced in the ecological literature-by Holling 6,7-as well as in the engineering literature-by Soliman and Thompson 8 (see 9 for a recent review)-who both argued that the local approach to stability needs to be accompanied by non-local measures capturing the characteristics of a state's basin of attraction. Recently, this global approach has been picked up in the field of complex networks in which multistability and large disturbances occur naturally as well 10-14. Suitable aspects of a basin of attraction are its size, shape and depth, which until now have all been used-separately 15-19 or in combination 20-22-to measure the stability of multistable systems. The suitability of a certain stability criterion for a specific problem also depends on the shape or distribution of the perturbations that occur. If perturbations are best described as noise, a suitable approach is to analyze the most likely escape path from the basin of attraction 23-25. This path is determined by the position of the saddle point on the basin boundary possessing the lowest barrier to escape 26. A very different situation occurs if perturbations are singular, large and abrupt. The response of nonlinear systems to such perturbations has been considered in terms of linear response theory as well as transient dynamics in climate science 27 , fluid dynamics 28,29 and energy networks 18,30,31. In networks, the most frequently applied characteristic to measure a system's stability against large abrupt perturbations is the relative basin size which is usually esti...

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Bounding the first exit from the basin: Independence times and finite-time basin stability

Cited by 21 publications

References 40 publications

Stochastic basins of attraction and generalized committor functions

Stochastic basins of attraction and generalized committor functions

Probabilistic Behavioral Distance and Tuning - Reducing and aggregating complex systems

Minimal fatal shocks in multistable complex networks

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