Dynamics and geometry near resonant bifurcations

Regular and Chaotic Dynamics

Volumn 16, Issue 1, 2011, Pages 39-50

  Broer, Henk W ^a   Holtman, Sijbo J ^a   Vegter, Gert ^a   Vitolo, Renato ^b  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

^b UNIVERSITY OF EXETER   (United Kingdom)

Author keywords

equivariant singularity theory; geometric structure; Lyapunov Schmidt reduction; periodically forced oscillator; resonant Hopf Ne marck Sacker bifurcation

Indexed keywords

EID: 79952024665     PISSN: 15603547     EISSN: 14684845     Source Type: Journal    
DOI: 10.1134/S1560354710520023     Document Type: Article

References (16)

1. Broer, H., Golubitsky, M., and Vegter, G., The Geometry of Resonance Tongues: A Singularity Approach, Nonlinearity, 2003, vol. 16, pp. 1511-1538.
2. Broer, H., Holtman, S., and Vegter, G., Recognition of the Bifurcation Type of Resonance in Mildly Degenerate Hopf-Neǐmark-Sacker Families, Nonlinearity, 2008, vol. 21, pp. 2463-2482.
3. Broer, H., Holtman, S., and Vegter, G., Recognition of Resonance Type in Periodically Forced Oscillators, Physica D, 2010, vol. 239, pp. 1627-1636.
4. Broer, H., Holtman, S., and Vegter, G., and Vitolo, R., Geometry and Dynamics of Mildly Degenerate Hopf-Neǐmarck-Sacker Families Near Resonance, Nonlinearity, 2009, vol. 22, pp. 2161-2200.
5. Arnol'd, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, Berlin: Springer, 1982.
6. Kuznetsov, Y., Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, vol. 112, Berlin, New-York: Springer, 1995.
7. Broer, H., Naudot, V., Roussarie, R., Saleh, K., and Wagener, F., Organising Centres in the Semi-global Analysis of Dynamical Systems, Int. J. Appl. M. Stat., 2007, vol. 12, no. D07, pp. 7-36.
8. 6-van der Pol Oscillator, Chaos, Solitons and Fractals, 2004, vol. 24, no. 4, pp. 841-853.
9. Duffing, G., Erzwungene schwingungen bei veränderlicher eigenfrequenz, Braunschweig: F. Vieweg u. Sohn 1918.
10. van der Pol, B. and van der Mark, J., Frequency Demultiplication, Nature, 1927, vol. 120, pp. 363-364.
11. Broer, H., Golubitsky, M., and Vegter, G., Geometry of Resonance Tongues, in Singularity Theory. Proceedings of the 2005 Marseille Singularity School and Conference, Hackensack, NJ: World Sci. Publ., 2007, pp. 327-356.
12. Broer, H., and Vegter, G., Generic Hopf-Neǐmarck-Sacker Bifurcations in Feed-forward Systems, Nonlinearity, 2008, vol. 21, pp. 1547-1578.
13. Takens, F., Forced Oscillations and Bifurcations, in Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973), Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974, Utrecht: Math. Inst. Rijksuniv. Utrecht, 1974, pp. 1-59.
14. Takens, F., Singularities of Vector Fields, Publ. Math. IHES, 1974, vol. 42, pp. 48-100.
15. Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, New York: Springer, 1983.
16. Vanderbauwhede, A., Branching of Periodic Solutions in Time-reversible Systems, in Geometry and Analysis in Non-linear Dynamics, H. Broer and F. Takens, Eds., Pitman Research Notes in Mathematics, vol. 222, Pitman London, 1992, pp. 97-113.