Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

Bram Lentjes; Len Spek; Maikel M. Bosschaert; Yuri A. Kuznetsov

doi:10.1016/j.jde.2025.01.064

Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

Bram Lentjes
, Len Spek
, Maikel M. Bosschaert
, Yuri A. Kuznetsov

Mathematics of Imaging & AI

Research output: Working paper › Preprint › Academic

148 Downloads (Pure)

Abstract

A recent work arXiv:2207.02480 by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates derivation of periodic normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that will allow us to describe the local dynamics on the center manifold in terms of these periodic normal forms. Furthermore, we characterize the center eigenspace by proving the existence of time periodic smooth Jordan chains for the original and the adjoint system.

Original language	English
Publisher	ArXiv.org
Number of pages	33
DOIs	https://doi.org/10.1016/j.jde.2025.01.064
Publication status	Published - 20 Feb 2023

Keywords

math.DS
math.FA
34C20, 34K13, 34K17, 34K18, 34K19

Access to Document

10.1016/j.jde.2025.01.064Licence: CC BY

2302.08806v1Final published version, 493 KBLicence: CC BY
2207.02480v2Final published version, 576 KB

Periodic normal forms for bifurcations of limit cycles in DDEs
Lentjes, B., Spek, L., Bosschaert, M. M. & Kuznetsov, Y. A., 5 Apr 2025, In: Journal of differential equations. 423, p. 631-694 64 p.
Research output: Contribution to journal › Article › Academic › peer-review

Open Access
File
2 Link opens in a new tab Citations (Scopus)

5 Downloads (Pure)

Cite this

@techreport{9ba2ad74a0cd4ef6abbbf830a36f3a88,

title = "Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs",

abstract = "A recent work arXiv:2207.02480 by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates derivation of periodic normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that will allow us to describe the local dynamics on the center manifold in terms of these periodic normal forms. Furthermore, we characterize the center eigenspace by proving the existence of time periodic smooth Jordan chains for the original and the adjoint system.",

keywords = "math.DS, math.FA, 34C20, 34K13, 34K17, 34K18, 34K19",

author = "Bram Lentjes and Len Spek and Bosschaert, \{Maikel M.\} and Kuznetsov, \{Yuri A.\}",

note = "33 pages, 1 figure",

year = "2023",

month = feb,

day = "20",

doi = "10.1016/j.jde.2025.01.064",

language = "English",

publisher = "ArXiv.org",

type = "WorkingPaper",

institution = "ArXiv.org",

}

TY - UNPB

T1 - Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

AU - Lentjes, Bram

AU - Spek, Len

AU - Bosschaert, Maikel M.

AU - Kuznetsov, Yuri A.

N1 - 33 pages, 1 figure

PY - 2023/2/20

Y1 - 2023/2/20

N2 - A recent work arXiv:2207.02480 by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates derivation of periodic normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that will allow us to describe the local dynamics on the center manifold in terms of these periodic normal forms. Furthermore, we characterize the center eigenspace by proving the existence of time periodic smooth Jordan chains for the original and the adjoint system.

AB - A recent work arXiv:2207.02480 by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates derivation of periodic normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that will allow us to describe the local dynamics on the center manifold in terms of these periodic normal forms. Furthermore, we characterize the center eigenspace by proving the existence of time periodic smooth Jordan chains for the original and the adjoint system.

KW - math.DS

KW - math.FA

KW - 34C20, 34K13, 34K17, 34K18, 34K19

U2 - 10.1016/j.jde.2025.01.064

DO - 10.1016/j.jde.2025.01.064

M3 - Preprint

BT - Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

PB - ArXiv.org

ER -

Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

Abstract

Keywords

Access to Document

Fingerprint

Research output

Periodic normal forms for bifurcations of limit cycles in DDEs

Cite this