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Quasi-linearization of parameter-depending germs of vector fields. (English) Zbl 1293.37017

Summary: In this paper, we study normalization and quasi-linearization of a family of germs of hyperbolic vector fields at the origin. We show that, when the eigenvalues of these systems satisfy the so-called strongly \(1\)-resonant condition, i.e., there exists a relation of the form \((r,\lambda)=0\), then they can be simplified, in the context of orbital equivalence, to a normal form which can be integrated in an explicit and convenient way. More precisely, given a family of germs of sufficiently smooth vector fields \(\mathcal{X}_\varepsilon\) having a generic nonlinear part, with \(\mathcal{X}_0\) strongly \(1\)-resonant, then for any integer \(k\), there is a neighbourhood \(\mathcal{V}\) at the origin of the parameter space such that for any \(\varepsilon\in\mathcal{V}\), \(\mathcal{X}_\varepsilon\) is \(C^k\) equivalent to a system having only one resonant term on each component.

MSC:

37C10 Dynamics induced by flows and semiflows
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
37G05 Normal forms for dynamical systems
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