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Journal of Dynamical and Control Systems

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24-10-2017

Normal Form for Second Order Differential Equations

Authors: Ilya Kossovskiy, D. Zaitsev

Published in: Journal of Dynamical and Control Systems | Issue 4/2018

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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).

previous article Zero Level Perturbation of a Certain Third-Order Linear Solvable ODE with an Irregular Singularity at the Origin of Poincaré Rank 1

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Title: Normal Form for Second Order Differential Equations
Authors: Ilya Kossovskiy
D. Zaitsev
Publication date: 24-10-2017
Publisher: Springer US
Published in: Journal of Dynamical and Control Systems / Issue 4/2018
Print ISSN: 1079-2724
Electronic ISSN: 1573-8698
DOI: https://doi.org/10.1007/s10883-017-9380-9

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