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On a theorem of Dulac. (Sur un théorème de Dulac.) (French) Zbl 0820.34023
Summary: We shall consider holomorphic vector fields of \(({\mathbb{C}}^ n,0)\) with a non zero diagonal linear part and which eigenvalues \(\lambda_ i\in{\mathbb{C}}\) satisfy to some resonance’s relations all generated by one relation \((r,\lambda)=0\) for a non zero vector \(r\in{\mathbb{N}}^ n\). We shall show that such vector fields can be transformed, by a local holomorphic diffeomorphism near \(0\in{\mathbb{C}}^ n\), into a preliminary normal form while exhibiting invariant varieties, if a hypothesis of diophantine small divisors is made. Our results generalize, to any dimension, those done by H. Dulac in dimension 2.

MSC:
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
34C40 Ordinary differential equations and systems on manifolds
34A34 Nonlinear ordinary differential equations and systems
37C80 Symmetries, equivariant dynamical systems (MSC2010)
37G05 Normal forms for dynamical systems
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