Smooth normalization of a vector field near a semistable limit cycle
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 Δ:( 1 ,0)( 1 ,0), Δ(x)=x+cx 2 +, 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.

     author = {Yakovenko, Sergey Yu.},
     title = {Smooth normalization of a vector field near a semistable limit cycle},
     journal = {Annales de l'Institut Fourier},
     pages = {893--903},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {3},
     year = {1993},
     doi = {10.5802/aif.1360},
     zbl = {0783.58071},
     mrnumber = {95a:58113},
     language = {en},
     url = {}
Yakovenko, Sergey Yu. Smooth normalization of a vector field near a semistable limit cycle. Annales de l'Institut Fourier, Tome 43 (1993) no. 3, pp. 893-903. doi : 10.5802/aif.1360.

[AVG] V.I. Arnold, A.N. Varchenko and S.M. Gussein-Zade, Singularities of differentiable maps. I. Classification of critical points, caustics and wavefronts, Monographs in Mathematics, vol 82, Birkhäuser Boston Inc., Boston MA, 1985. | Zbl 0554.58001

[B] G. Belitskiĭ, Equivalence and normal forms of germs of smooth mappings, Russian Mathematical Surveys, 33-1 (1978). | MR 80k:58017 | Zbl 0398.58009

[IY1] Yu.S. Ilyashenko, S.Yu. Yakovenko, Finite-differentiable normal forms for local families of diffeomorphisms and vector fields, Russian Math. Surveys, 46-1 (1991), 1-43. | Zbl 0744.58006

[IY2] Yu.S. Ilyashenko, S.Yu. Yakovenko, Nonlinear Stokes phenomena in smooth classification problems, Nonlinear Stokes phenomena (Yu. S. Ilyashenko, ed.), Advances in Soviet Mathematics, AMS Publ., Providence RI, 14 (1993), 235-287. | Zbl 0804.32012

[T] F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier, Grenoble, 23-2 (1973), 163-195. | Numdam | MR 51 #1872 | Zbl 0266.34046

[M] J.N. Mather, Stability of C∞-mappings, III, Publ. Math. IHES, 35 (1968), 279-308. | Numdam | Zbl 0159.25001