Random switching near bifurcations

Hurth, Tobias; Kuehn, Christian

doi:10.1142/s0219493720500082

Cited by 12 publications

(16 citation statements)

References 52 publications

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“…Notice the asymmetry in the switching rates that will lead to the system spending more and more time in the regime governed by u 0 as ε approaches 0. In [114], random switching between the vector fields U(•, p − , p − ) and U(•, p + , p + ) was studied in detail, for parameters p − < 0 and p + > 0 to the left and to the right of the transcritical bifurcation at p = 0. Even though the present setting is somewhat different, we will follow [114] quite closely.…”

Section: Piecewise Deterministic Processesmentioning

confidence: 99%

A General View on Double Limits in Differential Equations

Kuehn,

Berglund,

Bick

et al. 2021

Preprint

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In this paper, we review several results from singularly perturbed differential equations with multiple small parameters. In addition, we develop a general conceptual framework to compare and contrast the different results by proposing a three-step process. First, one specifies the setting and restrictions of the differential equation problem to be studied and identifies the relevant small parameters. Second, one defines a notion of equivalence via a property/observable for partitioning the parameter space into suitable regions near the singular limit. Third, one studies the possible asymptotic singular limit problems as well as perturbation results to complete the diagrammatic subdivision process. We illustrate this approach for two simple problems from algebra and analysis. Then we proceed to the review of several modern double-limit problems including multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each example, we illustrate the previously mentioned three-step process and show that already double-limit parametric diagrams provide an excellent unifying theme. After this review, we compare and contrast the common features among the different examples. We conclude with a brief outlook, how our methodology can help to systematize the field better, and how it can be transferred to a wide variety of other classes of differential equations.

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Section: Piecewise Deterministic Processesmentioning

confidence: 99%

A General View on Double Limits in Differential Equations

Kuehn,

Berglund,

Bick

et al. 2021

Preprint

Self Cite

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show abstract

“…Our method consists of looking at integrals of the function H = \scrL V with respect to invariant measures of the process. For some models, another method is possible, as used, for example, for some PDMPs in [22]. The idea is the following.…”

Section: 3mentioning

confidence: 99%

“…Assume that if \scrP inv (\scrM + ) is nonempty, then it is possible to compute, or at least, to estimate, the density of an invariant probability \mu \in \scrP inv (\scrM + ). Then, this density must satisfy some integrability conditions, which can be violated if \Lambda + (H) = 0 (see e.g [22,Theorem 3.1] or [17,Lemma 6]). Hence, if \Lambda + (H) = 0, \scrP inv (\scrM + ) has to be empty.…”

Section: 3mentioning

confidence: 99%

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A Method to Deal With the Critical Case in Stochastic Population Dynamics

Nguyen¹,

Strickler²

2020

SIAM J. Appl. Math.

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In numerous papers, the behavior of stochastic population models is investigated through the sign of a real quantity which is the growth rate of the population near the extinction set. In many cases, it is proven that when this growth rate is positive, the population is persistent in the long run, while if it is negative, the population goes extinct. However, the critical case when the growth rate is null is rarely treated. The aim of this paper is to provide a method that can be applied in many situations to prove that in the critical case, the process converges in temporal average to the extinction set. A number of applications are given for stochastic differential equations and piecewise deterministic Markov processes modeling prey-predator, epidemiological or structured population dynamics.

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“…Only very recently, the link between parametric uncertainty and bifurcations has emerged as a sub-area within the theory of nonlinear dynamics [2,12,15,16]. This link is important since parametric uncertainty leads to uncertainty in the location of tipping points and the shape of the bifurcation curve.…”

Section: Introductionmentioning

confidence: 99%

Uniting Parametric Uncertainty and Tipping Diagrams

Lux¹,

Ashwin²,

Wood³

et al. 2021

Preprint

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Various subsystems of the Earth system may undergo critical transitions by passing a socalled tipping point, under sustained changes to forcing. For example, the Atlantic Meridional Overturning Circulation (AMOC) is of particular importance for North Atlantic heat transport and is thought to be potentially at risk of tipping. Given a model of such a subsystem that accurately includes the relevant physical processes, whether tipping occurs or not, will depend on model parameters that typically are uncertain. Reducing this parametric uncertainty is important to understand the likelihood of tipping behavior being present in the system and possible tipping locations. In this letter, we develop improved estimates for the parametric uncertainty by inferring probability distributions for the model parameters based on physical constraints and by using a Bayesian inversion technique. To visualize the impact of parametric uncertainty, we extend classical tipping diagrams by visualizing probabilistic bifurcation curves according to the inferred distribution of the model parameter. Furthermore, we highlight the uncertain locations of tipping points along the probabilistic bifurcation curves. We showcase our probabilistic visualizations of the tipping behavior using a simple box-model of the AMOC, the Stommel-Cessi model [5].

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Random switching near bifurcations

Cited by 12 publications

References 52 publications

A General View on Double Limits in Differential Equations

A General View on Double Limits in Differential Equations

A Method to Deal With the Critical Case in Stochastic Population Dynamics

Uniting Parametric Uncertainty and Tipping Diagrams

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