Remarks on interacting Neimark-Sacker bifurcations

Yu A. Kuznetsov*, H. G.E. Meijer

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

27 Citations (Scopus)

Abstract

We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex eigenvalues together with either an eigenvalue - 1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher-order terms and these are investigated here. We (re)derive the normal forms and reduce them to a single amplitude map. This map is similar to the amplitude system for the double-Hopf bifurcation of vector fields. We show how the critical normal form coefficients determine the general bifurcation picture for this amplitude map. Representative nonsymmetric perturbations of the normal forms are studied numerically. Our case studies show a detailed picture near various bifurcation curves, which is richer than known theoretical predictions. For arbitrary maps with these bifurcations we give explicit formulas for critical normal form coefficients on center manifolds. We apply them to an example from robotics where we are able to demonstrate the existence of a bubble-structure, which was only observed in perturbed normal forms before.

Original languageEnglish
Pages (from-to)1009-1035
Number of pages27
JournalJournal of difference equations and applications
Volume12
Issue number10
DOIs
Publication statusPublished - 1 Oct 2006
Externally publishedYes

Fingerprint

Neimark-Sacker Bifurcation
Normal Form
Bifurcation
Hopf bifurcation
Bifurcation (mathematics)
Eigenvalue
Bifurcation Curve
Robotics
Center Manifold
Coefficient
Unfolding
Diffeomorphisms
Hopf Bifurcation
Bubble
Codimension
Explicit Formula
Vector Field
Fixed point
Higher Order
Perturbation

Keywords

  • Bifurcation
  • Diffeomorphisms
  • Nonsymmetric perturbation
  • Resonance

Cite this

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Remarks on interacting Neimark-Sacker bifurcations. / Kuznetsov, Yu A.; Meijer, H. G.E.

In: Journal of difference equations and applications, Vol. 12, No. 10, 01.10.2006, p. 1009-1035.

Research output: Contribution to journalArticleAcademicpeer-review

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T1 - Remarks on interacting Neimark-Sacker bifurcations

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AU - Meijer, H. G.E.

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N2 - We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex eigenvalues together with either an eigenvalue - 1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher-order terms and these are investigated here. We (re)derive the normal forms and reduce them to a single amplitude map. This map is similar to the amplitude system for the double-Hopf bifurcation of vector fields. We show how the critical normal form coefficients determine the general bifurcation picture for this amplitude map. Representative nonsymmetric perturbations of the normal forms are studied numerically. Our case studies show a detailed picture near various bifurcation curves, which is richer than known theoretical predictions. For arbitrary maps with these bifurcations we give explicit formulas for critical normal form coefficients on center manifolds. We apply them to an example from robotics where we are able to demonstrate the existence of a bubble-structure, which was only observed in perturbed normal forms before.

AB - We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex eigenvalues together with either an eigenvalue - 1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher-order terms and these are investigated here. We (re)derive the normal forms and reduce them to a single amplitude map. This map is similar to the amplitude system for the double-Hopf bifurcation of vector fields. We show how the critical normal form coefficients determine the general bifurcation picture for this amplitude map. Representative nonsymmetric perturbations of the normal forms are studied numerically. Our case studies show a detailed picture near various bifurcation curves, which is richer than known theoretical predictions. For arbitrary maps with these bifurcations we give explicit formulas for critical normal form coefficients on center manifolds. We apply them to an example from robotics where we are able to demonstrate the existence of a bubble-structure, which was only observed in perturbed normal forms before.

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