A general view on double limits in differential equations

Christian Kuehn; Nils Berglund; Christian Bick; Maximilian Engel; Tobias Hurth; Annalisa Iuorio; Cinzia Soresina

doi:10.1016/j.physd.2021.133105

A general view on double limits in differential equations

Christian Kuehn^*, Nils Berglund, Christian Bick, Maximilian Engel, Tobias Hurth, Annalisa Iuorio, Cinzia Soresina

^*Corresponding author for this work

Research output: Contribution to Journal › Review article › Academic › peer-review

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Abstract

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.

Original language	English
Article number	133105
Pages (from-to)	1-26
Number of pages	26
Journal	Physica D: Nonlinear Phenomena
Volume	431
Early online date	1 Dec 2021
DOIs	https://doi.org/10.1016/j.physd.2021.133105
Publication status	Published - Mar 2022

Bibliographical note

Funding Information:
CK has been supported by a Lichtenberg Professorship of the VolkswagenStiftung. CK also acknowledges inspiring discussions with Grigorios A. Pavliotis regarding limit problems in differential equations, which were made possible by a TUM John von Neumann Visiting Professorship. NB has been supported by the ANR project PERISTOCH, ANR?19?CE40?0023. CB has been supported by the Institute for Advanced Study at the Technical University of Munich, Germany through a Hans Fischer fellowship and the Engineering and Physical Sciences Research Council (EPSRC), UK through the grant EP/T013613/1. ME has been supported by Germany's Excellence Strategy ? The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689). AI acknowledges support by an FWF Hertha Firnberg Research Fellowship (T 1199-N). CS has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk?odowska?Curie grant agreement No. 754462. Support by INdAM-GNFM is gratefully acknowledged by CS. TH gratefully acknowledges support through SNF grant 200021?175728/1. We also thank two anonymous referees, whose comments and suggestions have helped to improve the presentation of this work.

Funding Information:
CK has been supported by a Lichtenberg Professorship of the VolkswagenStiftung. CK also acknowledges inspiring discussions with Grigorios A. Pavliotis regarding limit problems in differential equations, which were made possible by a TUM John von Neumann Visiting Professorship. NB has been supported by the ANR project PERISTOCH , ANR–19–CE40–0023 . CB has been supported by the Institute for Advanced Study at the Technical University of Munich, Germany through a Hans Fischer fellowship and the Engineering and Physical Sciences Research Council (EPSRC), UK through the grant EP/T013613/1 . ME has been supported by Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689 ). AI acknowledges support by an FWF Hertha Firnberg Research Fellowship ( T 1199-N ). CS has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska–Curie grant agreement No. 754462 . Support by INdAM-GNFM is gratefully acknowledged by CS. TH gratefully acknowledges support through SNF grant . We also thank two anonymous referees, whose comments and suggestions have helped to improve the presentation of this work.

Publisher Copyright:
© 2021 Elsevier B.V.

Funding

CK has been supported by a Lichtenberg Professorship of the VolkswagenStiftung. CK also acknowledges inspiring discussions with Grigorios A. Pavliotis regarding limit problems in differential equations, which were made possible by a TUM John von Neumann Visiting Professorship. NB has been supported by the ANR project PERISTOCH, ANR?19?CE40?0023. CB has been supported by the Institute for Advanced Study at the Technical University of Munich, Germany through a Hans Fischer fellowship and the Engineering and Physical Sciences Research Council (EPSRC), UK through the grant EP/T013613/1. ME has been supported by Germany's Excellence Strategy ? The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689). AI acknowledges support by an FWF Hertha Firnberg Research Fellowship (T 1199-N). CS has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk?odowska?Curie grant agreement No. 754462. Support by INdAM-GNFM is gratefully acknowledged by CS. TH gratefully acknowledges support through SNF grant 200021?175728/1. We also thank two anonymous referees, whose comments and suggestions have helped to improve the presentation of this work. CK has been supported by a Lichtenberg Professorship of the VolkswagenStiftung. CK also acknowledges inspiring discussions with Grigorios A. Pavliotis regarding limit problems in differential equations, which were made possible by a TUM John von Neumann Visiting Professorship. NB has been supported by the ANR project PERISTOCH , ANR–19–CE40–0023 . CB has been supported by the Institute for Advanced Study at the Technical University of Munich, Germany through a Hans Fischer fellowship and the Engineering and Physical Sciences Research Council (EPSRC), UK through the grant EP/T013613/1 . ME has been supported by Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689 ). AI acknowledges support by an FWF Hertha Firnberg Research Fellowship ( T 1199-N ). CS has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska–Curie grant agreement No. 754462 . Support by INdAM-GNFM is gratefully acknowledged by CS. TH gratefully acknowledges support through SNF grant . We also thank two anonymous referees, whose comments and suggestions have helped to improve the presentation of this work.

Funders	Funder number
INdAM-GNFM
Horizon 2020 Framework Programme
Engineering and Physical Sciences Research Council	EP/T013613/1, EXC-2046/1, 390685689
Volkswagen Foundation
Agence Nationale de la Recherche
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung	200021−175728/1
Austrian Science Fund	T 1199-N
Horizon 2020	754462
Institute for Advanced Study, Technische Universität München

Keywords

Double limits
Multiscale dynamics
Singular perturbation
Stochastic dynamics

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10.1016/j.physd.2021.133105

A general view on double limits in differential equationsFinal published version, 1.33 MBLicence: Article 25fa Dutch Copyright Act

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title = "A general view on double limits in differential equations",

abstract = "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.",

keywords = "Double limits, Multiscale dynamics, Singular perturbation, Stochastic dynamics",

author = "Christian Kuehn and Nils Berglund and Christian Bick and Maximilian Engel and Tobias Hurth and Annalisa Iuorio and Cinzia Soresina",

note = "Funding Information: CK has been supported by a Lichtenberg Professorship of the VolkswagenStiftung. CK also acknowledges inspiring discussions with Grigorios A. Pavliotis regarding limit problems in differential equations, which were made possible by a TUM John von Neumann Visiting Professorship. NB has been supported by the ANR project PERISTOCH, ANR?19?CE40?0023. CB has been supported by the Institute for Advanced Study at the Technical University of Munich, Germany through a Hans Fischer fellowship and the Engineering and Physical Sciences Research Council (EPSRC), UK through the grant EP/T013613/1. ME has been supported by Germany's Excellence Strategy ? The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689). AI acknowledges support by an FWF Hertha Firnberg Research Fellowship (T 1199-N). CS has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk?odowska?Curie grant agreement No. 754462. Support by INdAM-GNFM is gratefully acknowledged by CS. TH gratefully acknowledges support through SNF grant 200021?175728/1. We also thank two anonymous referees, whose comments and suggestions have helped to improve the presentation of this work. Funding Information: CK has been supported by a Lichtenberg Professorship of the VolkswagenStiftung. CK also acknowledges inspiring discussions with Grigorios A. Pavliotis regarding limit problems in differential equations, which were made possible by a TUM John von Neumann Visiting Professorship. NB has been supported by the ANR project PERISTOCH , ANR–19–CE40–0023 . CB has been supported by the Institute for Advanced Study at the Technical University of Munich, Germany through a Hans Fischer fellowship and the Engineering and Physical Sciences Research Council (EPSRC), UK through the grant EP/T013613/1 . ME has been supported by Germany{\textquoteright}s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689 ). AI acknowledges support by an FWF Hertha Firnberg Research Fellowship ( T 1199-N ). CS has received funding from the European Union{\textquoteright}s Horizon 2020 research and innovation program under the Marie Sk{\l}odowska–Curie grant agreement No. 754462 . Support by INdAM-GNFM is gratefully acknowledged by CS. TH gratefully acknowledges support through SNF grant . We also thank two anonymous referees, whose comments and suggestions have helped to improve the presentation of this work. Publisher Copyright: {\textcopyright} 2021 Elsevier B.V.",

year = "2022",

month = mar,

doi = "10.1016/j.physd.2021.133105",

language = "English",

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AU - Bick, Christian

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AU - Iuorio, Annalisa

AU - Soresina, Cinzia

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N2 - 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.

AB - 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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KW - Multiscale dynamics

KW - Singular perturbation

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A general view on double limits in differential equations

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