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Symmetries of center singularities of plane vector fields

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Nonlinear Oscillations

Let \( {D^2} \subset {\mathbb{R}^2} \) be a closed unit 2-disk centered at the origin \( O \in {\mathbb{R}^2} \) and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let \( \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) \) be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also \( {\mathcal{D}^{+} }(F) \) be the group of diffeomorphisms of D 2 that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D 2 if and only if the 1-jet of F at O is a “rotation,” i.e., \( {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} \). Then \( {\mathcal{D}^{+} }(F) \) is homotopy equivalent to a circle.

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  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    S. I. Maksymenko

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Correspondence to S. I. Maksymenko.

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Published in Neliniini Kolyvannya, Vol. 13, No. 2, pp. 177–205, April–June, 2010.

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Maksymenko, S.I. Symmetries of center singularities of plane vector fields. Nonlinear Oscill 13, 196–227 (2010). https://doi.org/10.1007/s11072-010-0110-4

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