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    First Integrals and Normal Forms for Germs of Analytic Vector Fields

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    2006-411.pdf (274.9Kb)
    Date
    2006
    Author
    Chen, Jian
    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.
    URI
    http://hdl.handle.net/1853/29409
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    • School of Mathematics Faculty Publications [119]

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