Smooth normalization of a vector field near a semistable limit cycle

Yakovenko, Sergey Yu.. "Smooth normalization of a vector field near a semistable limit cycle." Annales de l'institut Fourier 43.3 (1993): 893-903. <http://eudml.org/doc/75024>.

@article{Yakovenko1993,
abstract = {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 $\Delta :(\{\Bbb R\}^1,0) \rightarrow (\{\Bbb R\}^1,0)$, $\Delta (x)=x+cx^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.},
author = {Yakovenko, Sergey Yu.},
journal = {Annales de l'institut Fourier},
keywords = {smooth normal forms; homotopy method; homological equation; limit cycle},
language = {eng},
number = {3},
pages = {893-903},
publisher = {Association des Annales de l'Institut Fourier},
title = {Smooth normalization of a vector field near a semistable limit cycle},
url = {http://eudml.org/doc/75024},
volume = {43},
year = {1993},
}

TY - JOUR
AU - Yakovenko, Sergey Yu.
TI - Smooth normalization of a vector field near a semistable limit cycle
JO - Annales de l'institut Fourier
PY - 1993
PB - Association des Annales de l'Institut Fourier
VL - 43
IS - 3
SP - 893
EP - 903
AB - 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 $\Delta :({\Bbb R}^1,0) \rightarrow ({\Bbb R}^1,0)$, $\Delta (x)=x+cx^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.
LA - eng
KW - smooth normal forms; homotopy method; homological equation; limit cycle
UR - http://eudml.org/doc/75024
ER -