Continuation of bifurcations in periodic delay-differential equations using characteristic matrices

Róbert Szalai, G. Stépán, S. John Hogan

Research output: Contribution to journalArticle

42 Citations (Scopus)

Abstract

In this paper we describe a method for continuing periodic solution bifurcations in periodic delay-differential equations. First, the notion of characteristic matrices of periodic orbits is introduced and equivalence with the monodromy operator is demonstrated. An alternative formulation of the characteristic matrix is given, which can be computed efficiently. Defining systems of bifurcations are constructed in a standard way, including the characteristic matrix and its derivatives. For following bifurcation curves in two parameters, the pseudo-arclength method is used combined with Newton iteration. Two test examples (an interrupted machining model and a traffic model with driver reaction time) conclude the paper. The algorithm has been implemented in the software tool PDDE-CONT.

Original languageEnglish
Pages (from-to)1301-1317
Number of pages17
JournalSIAM Journal on Scientific Computing
Volume28
Issue number4
DOIs
Publication statusPublished - 2006

Fingerprint

Bifurcation (mathematics)
Delay Differential Equations
Continuation
Differential equations
Bifurcation
Bifurcation Curve
Newton Iteration
Reaction Time
Traffic Model
Monodromy
Software Tools
Machining
Periodic Orbits
Driver
Two Parameters
Periodic Solution
Orbits
Equivalence
Derivatives
Derivative

Keywords

  • Bifurcations
  • Continuation
  • Delay-differential equations
  • Periodic solutions

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Continuation of bifurcations in periodic delay-differential equations using characteristic matrices. / Szalai, Róbert; Stépán, G.; Hogan, S. John.

In: SIAM Journal on Scientific Computing, Vol. 28, No. 4, 2006, p. 1301-1317.

Research output: Contribution to journalArticle

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