Smooth normalization of a vector field near a semistable limit cycle

Yakovenko, Sergey Yu.

Annales de l'Institut Fourier, Tome 43 (1993) no. 3, pp. 893-903.

Nous montrons qu’un champ de vecteurs différentiable admet une forme normale polynomiale intégrable autour d’un cycle limite de multiplicité deux. Cette forme dépend de trois paramètres : l’invariant formel de monodromie, la période du cycle et un troisième invariant qui mesure l’asymétrie des périodes des cycles apparaissent dans la bifurcation générique de ce cycle double.

We establish a polynomial normal form for a vector field having a limit cycle of multiplicity 2. The smooth classification problem for such fields is closely related to the problem of classification of germs $\Delta :({ℝ}^1,0)\to ({ℝ}^1,0)$, $\Delta \left(x\right)=x+c{x}^2+\cdots$, solved by F. Takens in 1973. Such germs appear as the germs of Poincaré return maps for semistable cycles, and a smooth conjugacy between any two such germs may be extended to a smooth orbital equivalence between the original fields.

If one deals with smooth conjugacy of flows rather than with the orbital equivalence of the corresponding fields, then two additional real parameters appear. One of them is the period of the cycle, while the second parameter keeps track of the asymmetry of the angular velocity, resulting in a difference between periods of two hyperbolic cycles appearing after perturbation of the given field.

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