Naudot, Vincent; Yang, Jiazhong Quasi-linearization of parameter-depending germs of vector fields. (English) Zbl 1293.37017 Dyn. Syst. 28, No. 2, 173-186 (2013). 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 Keywords:quasi-linearization; vector fields; integrability; normal form PDFBibTeX XMLCite \textit{V. Naudot} and \textit{J. Yang}, Dyn. Syst. 28, No. 2, 173--186 (2013; Zbl 1293.37017) Full Text: DOI References: [1] DOI: 10.1007/978-3-662-02535-2 · doi:10.1007/978-3-662-02535-2 [2] DOI: 10.1080/14689360802331162 · Zbl 1159.34325 · doi:10.1080/14689360802331162 [3] Samovol V S, Proc Moscow Math Soc 38 pp 187– (1979) [4] DOI: 10.1007/BF01159105 · Zbl 0715.34074 · doi:10.1007/BF01159105 [5] DOI: 10.1142/S0218127406016951 · Zbl 1117.37020 · doi:10.1142/S0218127406016951 [6] DOI: 10.1017/S014338570001018X · Zbl 0866.58045 · doi:10.1017/S014338570001018X [7] DOI: 10.3934/dcdss.2010.3.667 · Zbl 1216.37014 · doi:10.3934/dcdss.2010.3.667 [8] DOI: 10.1215/S0012-7094-01-10611-X · Zbl 1020.34033 · doi:10.1215/S0012-7094-01-10611-X [9] DOI: 10.1070/RM1991v046n01ABEH002733 · Zbl 0744.58006 · doi:10.1070/RM1991v046n01ABEH002733 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.