Periodic delay orbits and the polyfold implicit function theorem
We consider differential delay equations of the form partial derivative(t)x(t) = X-t(x(t-tau)) in R-n, where (X-t)(t is an element of S)1 is a time-dependent family of smooth vector fields on R-n and tau is a delay parameter. If there is a (suitably non-degenerate) periodic solution x(0) of this equ...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article (Journal) |
| Language: | English |
| Published: |
7 July 2022
|
| In: |
Commentarii mathematici Helvetici
Year: 2022, Volume: 97, Issue: 2, Pages: 383-412 |
| ISSN: | 1420-8946 |
| DOI: | 10.4171/CMH/533 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.4171/CMH/533 Verlag, lizenzpflichtig, Volltext: https://www.webofscience.com/api/gateway?GWVersion=2&SrcAuth=DynamicDOIArticle&SrcApp=WOS&KeyAID=10.4171%2FCMH%2F533&DestApp=DOI&SrcAppSID=EUW1ED0C98ANz1hFAQwiiDBqGFa2p&SrcJTitle=COMMENTARII+MATHEMATICI+HELVETICI&DestDOIRegistrantName=EMS+Press 
|
| Author Notes: | Peter Albers, Irene Seifert |
| Summary: | We consider differential delay equations of the form partial derivative(t)x(t) = X-t(x(t-tau)) in R-n, where (X-t)(t is an element of S)1 is a time-dependent family of smooth vector fields on R-n and tau is a delay parameter. If there is a (suitably non-degenerate) periodic solution x(0) of this equation for tau = 0, that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by tau. However, it seems difficult to prove this using the classical implicit function theorem, since the equation above, considered as an operator, is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer-Wysocki-Zehnder (2009, 2021) to overcome this problem in a natural setup. |
|---|---|
| Item Description: | Gesehen am 11.08.2022 |
| Physical Description: | Online Resource |
| ISSN: | 1420-8946 |
| DOI: | 10.4171/CMH/533 |