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Normal Form for Second Order Differential Equations

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Abstract

Applying methods of CR-geometry, we give a solution to the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a complete normal form (which is smooth or analytic respectively) for this class of ordinary differential equations (ODEs). The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE y ″ = 0. For a generic ODE, we also provide a unique (up to a discrete group action) normal form. By doing so, we give a solution to a problem which remained unsolved since the work of Arnold (1988). As another application of the normal form, we obtain distinguished curves associated with a differential equation that we call chains due to their analogy with the chains defined by Chern and Moser (Acta Math. 7;133:219–271).

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Acknowledgements

The authors are sincerely grateful to Sergei Yakovenko for multiple useful discussions held during the preparation of the paper and his valuable suggestions. We also thank Boris Kruglikov for his useful comments on the text. The first author is thankful to Alexander Sukhov for inspiring discussions towards the normal form problem. The first author is supported by the Czech Grant Agency and the Austrian Science Fund (FWF).

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Ilya Kossovskiy

  2. Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic

    Ilya Kossovskiy

  3. School of Mathematics, Trinity College Dublin, Dublin, Ireland

    D. Zaitsev

Authors
  1. Ilya Kossovskiy
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  2. D. Zaitsev
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Correspondence to Ilya Kossovskiy.

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Kossovskiy, I., Zaitsev, D. Normal Form for Second Order Differential Equations. J Dyn Control Syst 24, 541–562 (2018). https://doi.org/10.1007/s10883-017-9380-9

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  • DOI: https://doi.org/10.1007/s10883-017-9380-9

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