The geometry of Chazy's third order homogeneous differential equations

Details The geometry of Chazy's third order homogeneous differential equations The geometry of Chazy's third order homogeneous differential equations

2010-11-28

arxiv.org/abs/1011.6090v1

Mathematics

Dynamical Systems

16 pages

Scientific paper

Chazy studied a family of homogeneous third-order autonomous differential equations. They are those, within a certain class, admitting exclusively single-valued solutions. Each one of these equations yields a polynomial vector field in $\mathbf{C}^3$. For almost all of these these vector fields, the Zariski closure of a generic orbit yields an affine surface endowed with a holomorphic vector field that has exclusively single-valued solutions. We classify these surfaces and relate this classification to recent results of Rebelo and the author.

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