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Resonance and Singularities

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Progress and Challenges in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 54))

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Abstract

The phenomenon of resonance will be dealt with from the viewpoint of dynamical systems depending on parameters and their bifurcations. Resonance phenomena are associated to open subsets in the parameter space, while their complement corresponds to quasi-periodicity and chaos. The latter phenomena occur for parameter values in fractal sets of positive measure. We describe a universal phenomenon that plays an important role in modelling. This paper gives a summary of the background theory, veined by examples.

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Notes

  1. 1.

    For simplicity we take \(\vert \varepsilon \vert < 1\) which ensures that (9) is a circle diffeomorphism; for \(\vert \varepsilon \vert \geq 1\) the mapping becomes a circle endomorphism and the current approach breaks down.

  2. 2.

    Mathematical ideas on adiabatic change were used earlier by Rayleigh and Poincaré and by Landau-Lifschitz.

  3. 3.

    For ‘historical’ reasons we use the letter a instead of α2 as we did earlier.

  4. 4.

    The co-ordinates (ζ,t) sometimes also are called co-rotating. Also think of the Lagrangean variation of constants.

  5. 5.

    Colloquially often referred to as ‘chaotic sea’.

  6. 6.

    Also known as the metric entropy conjecture.

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Acknowledgements

The author thanks Konstantinos Efstathiou, Aernout van Enter and Ferdinand Verhulst for their help in the preparation of this paper.

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  1. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, 407, 9700 AK, Groningen, The Netherlands

    Henk W. Broer & Gert Vegter

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  1. Henk W. Broer
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  1. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Santiago Ibáñez

  2. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Jesús S. Pérez del Río

  3. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Antonio Pumariño

  4. Department of Mathematics, University of Oviedo, Oviedo, Spain

    J. Ángel Rodríguez

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Broer, H.W., Vegter, G. (2013). Resonance and Singularities. In: Ibáñez, S., Pérez del Río, J., Pumariño, A., Rodríguez, J. (eds) Progress and Challenges in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38830-9_7

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