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V-Shaped Action Functional with Delay

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Abstract

In this note we introduce the V-shaped action functional with delay in a symplectization, which is an intermediate action functional between the Rabinowitz action functional and the V-shaped action functional. It lives on the same space as the V-shaped action functional, but its gradient flow equation is a delay equation as in the case of the Rabinowitz action functional. We show that there is a smooth interpolation between the V-shaped action functional and the V-shaped action functional with delay during which the critical points and its actions are fixed. Moreover, we prove that there is a bijection between gradient flow lines of the V-shaped action functional with delay and the ones of the Rabinowitz action functional.

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Change history

  • 29 October 2023

    The numbering issue has been changed to 4-5.

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ACKNOWLEDGMENTS

The author would like to thank Felix Schlenk for extremely useful comments.

Funding

The author acknowledges partial support by DFG grant FR 2637/2-2.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Augsburg, Universitätsstraße 14, 86159, Augsburg, Germany

    Urs Frauenfelder

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  1. Urs Frauenfelder
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Correspondence to Urs Frauenfelder.

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The author declares that he has no conflicts of interest.

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MSC2010

37J39, 57R58

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Frauenfelder, U. V-Shaped Action Functional with Delay. Regul. Chaot. Dyn. 28, 364–373 (2023). https://doi.org/10.1134/S1560354723510020

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  • DOI: https://doi.org/10.1134/S1560354723510020

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