Title:
First Integrals and Normal Forms for Germs of Analytic Vector Fields
First Integrals and Normal Forms for Germs of Analytic Vector Fields
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Author(s)
Chen, Jian
Yi, Yingfei
Zhang, Xiang
Yi, Yingfei
Zhang, Xiang
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Abstract
For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant
relations for a quasi-periodic vector field, which generalizes one of the Poincaré’s classical results [18] on autonomous systems and Theorem 5 of [14] on periodic systems. Then in the space of analytic autonomous systems in C[2n] with exactly n resonances and n functionally independent first
integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler-Lagrange
equation we provide a new approach to its proof.
Sponsor
The second author was partially supported by NSF grant DMS0204119. The third author is partially supported by NSFC grant 10231020, Shuguang plan of Shanghai grant 03SG10 and NCET.
Date Issued
2006
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Pre-print