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Regular and Chaotic Dynamics, 1999, Volume 4, Issue 2, Pages 16–43
DOI: https://doi.org/10.1070/RD1999v004n02ABEH000103 (Mi rcd900) 		 
This article is cited in 12 scientific papers (total in 12 papers)

Integrable and non-integrable deformations of the skew Hopf bifurcation

H. W. Broera, F. Takensa, F. O. O. Wagenerb

a University of Groningen, Department of Mathematics, P.O. Box 800, 9700 AV Groningen, Netherlands
b Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
Citations (12)
DOI: https://doi.org/10.1070/RD1999v004n02ABEH000103
Abstract: In the skew Hopf bifurcation a quasi-periodic attractor with nontrivial normal linear dynamics loses hyperbolicity. Periodic, quasi-periodic and chaotic dynamics occur, including motion with mixed spectrum. The case of 3-dimensional skew Hopf bifurcation families of diffeomorphisms near integrability is discussed, surveying some recent results in a broad perspective. One result, using KAM-theory, deals with the persistence of quasi-periodic circles. Other results concern the bifurcations of periodic attractors in the case of resonance.
Received: 29.07.1999
Bibliographic databases:
Document Type: Article
MSC: 34C15, 34C20, 58F27, 70H05
Language: English
Citation: H. W. Broer, F. Takens, F. O. O. Wagener, “Integrable and non-integrable deformations of the skew Hopf bifurcation”, Regul. Chaotic Dyn., 4:2 (1999), 16–43
Citation in format AMSBIB
\Bibitem{BroTakWag99}
\by H.~W.~Broer, F. Takens, F. O. O. Wagener
\paper Integrable and non-integrable deformations of the skew Hopf bifurcation
\jour Regul. Chaotic Dyn.
\yr 1999
\vol 4
\issue 2
\pages 16--43
\mathnet{http://mi.mathnet.ru/rcd900}
\crossref{https://doi.org/10.1070/RD1999v004n02ABEH000103}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1781156}
\zmath{https://zbmath.org/?q=an:1012.37031}
Linking options:
  • https://www.mathnet.ru/eng/rcd900
  • https://www.mathnet.ru/eng/rcd/v4/i2/p16
    • This publication is cited in the following 12 articles:
    1. Henk W. Broer, Heinz Hanßmann, Florian Wagener, “Parametrised KAM Theory, an Overview”, Regul. Chaot. Dyn., 2025  crossref
    2. Yuri A. Kuznetsov, Hil G. E. Meijer, Numerical Bifurcation Analysis of Maps, 2019  crossref
    3. Henk W. Broer, Mathematics of Complexity and Dynamical Systems, 2012, 1152  crossref
    4. Renato Vitolo, Henk Broer, Carles Simó, “Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems”, Regul. Chaotic Dyn., 16:1 (2011), 154–184  mathnet  crossref
    5. H.W. Broer, Mikhail B. Sevryuk, Handbook of Dynamical Systems, 3, 2010, 249  crossref
    6. Renato Vitolo, Henk Broer, Carles Simó, “Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms”, Nonlinearity, 23:8 (2010), 1919  crossref
    7. Henk W. Broer, Encyclopedia of Complexity and Systems Science, 2009, 6310  crossref
    8. Henk W. Broer, Encyclopedia of Complexity and Systems Science Series, Perturbation Theory, 2009, 79  crossref
    9. Henk Broer, Carles Simó, Renato Vitolo, “Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance 'bubble'”, Physica D: Nonlinear Phenomena, 237:13 (2008), 1773  crossref
    10. Yu. A. Kuznetsov, H. G. E. Meijer, “Remarks on interacting Neimark–Sacker bifurcations”, Journal of Difference Equations and Applications, 12:10 (2006), 1009  crossref
    11. Henk W Broer, Heinz Hanßmann, Jiangong You, “Bifurcations of normally parabolic tori in Hamiltonian systems”, Nonlinearity, 18:4 (2005), 1735  crossref
    12. F Takens, F O O Wagener, “Resonances in skew and reducible quasi-periodic Hopf bifurcations”, Nonlinearity, 13:2 (2000), 377  crossref
    Citing articles in Google Scholar: Russian citations, English citations
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