Journal of Dynamical and Control Systems

, Volume 10, Issue 4, pp 501–525 | Cite as

Normalizability, Synchronicity, and Relative Exactness for Vector Fields in C2

  • C. Christopher
  • P. Mardešić
  • C. Rousseau


In this paper, we study the necessary and su.cient condition under which an orbitally normalizable vector field of saddle or saddle-node type in C2is analytically conjugate to its formal normal form (i.e., normalizable) by a transformation fixing the leaves of the foliation locally.

First, we express this condition in terms of the relative exactness of a certain 1-form derived from comparing the time-form of the vector field with the time-form of the normal form. Then we show that this condition is equivalent to a synchronicity condition: the vanishing of the integral of this 1-form along certain asymptotic cycles de.ned by the vector field. This can be seen as a generalization of the classical Poincaré theorem saying that a center is isochronous (i.e., synchronous to the linear center) if and only if it is linearizable.

The results, in fact, allow us in many cases to compare any two vector fields which differ by a multiplicative factor. In these cases we show that the two vector fields are analytically conjugate by a transformation fixing the leaves of the foliation locally if and only if their time-forms are synchronous.


Vector Field System Theory Normal Form Normalizable Vector Multiplicative Factor 
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Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • C. Christopher
    • 1
  • P. Mardešić
    • 2
  • C. Rousseau
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of PlymouthPlymouth, DevonU.K
  2. 2.Laboratoire de Topologie, Unité mixte de recherche 5584 du C.N.R.S.Université de BourgogneDijon CedexFrance
  3. 3.Département de mathématiques et de statistique and CRMUniversité de MontréalQuébecCanada

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