We study the problem of the existence and of the geometric structure of the set of periodic orbits of a vector field in presence of a first integral. We give a unified treatment and a geometric proof of existence results of periodic orbits by Moser (local case) and Bottkol (global case) under a suitable nonresonance condition.The local resonance case is considered, too. For analytic vector fields admitting an analytic first integral, we give a geometric description of the set of periodic orbits, proving that it is an analytic set, hence extending a theorem by Siegel.
A geometric approach to the existence of sets of periodic orbits / Margheri, A; Villarini, Massimo. - In: JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS. - ISSN 1040-7294. - STAMPA. - 14:(2002), pp. 835-853. [10.1023/A:1020764611067]
A geometric approach to the existence of sets of periodic orbits
VILLARINI, Massimo
2002
Abstract
We study the problem of the existence and of the geometric structure of the set of periodic orbits of a vector field in presence of a first integral. We give a unified treatment and a geometric proof of existence results of periodic orbits by Moser (local case) and Bottkol (global case) under a suitable nonresonance condition.The local resonance case is considered, too. For analytic vector fields admitting an analytic first integral, we give a geometric description of the set of periodic orbits, proving that it is an analytic set, hence extending a theorem by Siegel.
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