Singularities of invariant densities for random switching between two linear ODEs in 2D

Bakhtin, Yuri; Hurth, Tobias; Lawley, Sean D.; Mattingly, Jonathan C.

doi:10.48550/arxiv.2009.01299

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Piecewise Deterministic Processes2

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Cited by 2 publications

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“…For switching systems in dimension greater than one, the set of singularities of invariant densities can have a much richer structure than the one exhibited here (see [122] for a simple yet nontrivial example in 2D). This can result in more complex classification diagrams with respect to a suitably defined version of P bdd .…”

Section: Piecewise Deterministic Processesmentioning

confidence: 79%

A general view on double limits in differential equations

Kuehn

Berglund

Bick

et al. 2022

Physica D: Nonlinear Phenomena

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Section: Piecewise Deterministic Processesmentioning

confidence: 79%

A general view on double limits in differential equations

Kuehn

Berglund

Bick

et al. 2022

Physica D: Nonlinear Phenomena

Self Cite

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“…For switching systems in dimension greater than one, the set of singularities of invariant densities can have a much richer structure than the one exhibited here (see [10] for a simple yet nontrivial example in 2D). This can result in more complex classification diagrams with respect to a suitably defined version of P bdd .…”

Section: Piecewise Deterministic Processesmentioning

confidence: 79%

A General View on Double Limits in Differential Equations

Kuehn,

Berglund,

Bick

et al. 2021

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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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Singularities of invariant densities for random switching between two linear ODEs in 2D

Cited by 2 publications

References 18 publications

A general view on double limits in differential equations

A general view on double limits in differential equations

A General View on Double Limits in Differential Equations

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