Naslov
Normalizability, synchronicity, and relative exactness for vector fields in C2

Autori
Christopher, C. ; Mardešić, Pavao ; Rousseau, C.

Izvornik
Journal of dynamical and control systems (1079-2724) 10 (2004), 4; 501-525

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
normalizability; synchronicity

Sažetak
In this paper we study the necessary and sufficient condition under which an orbitally normalizable vector field of saddle or saddle-node type in C 2 is analyt- ically conjugate to its formal normal form (i.e. normalizable) by a trans formation fixing the leaves of the foliation locally. We first 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. We then show that this condition is equivalent to a synchronicity condition : the vanishing of the integral of this 1-form along certain asymptotic cycles defined by the vector field. This can be seen as a generalization of the classical theorem of Poincar ́e 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 foliat ion locally if and only if their time-forms are synchronous.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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