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Normalizable, integrable, and linearizable saddle points for complex quadratic systems in ${ℂ}^{2}$. (English) Zbl 1022.37035

In this paper, the authors consider complex differential systems in the neighborhood of a singular point with eigenvalues in the ratio $1:-\lambda$ with a real positive $\lambda ·$ They address the questions of orbital normalizability, normalizability (i.e., convergence of the normalizing transformation), integrability (i.e., orbital linearizability), and linearizability of the system. As for the experimental part of their study, the authors specialize to quadratic systems and study the values of $\lambda$ for which these notions are distinct. For this purpose they give several tools for demonstrating normalizability, integrability, and linearizability.

The authors’ main interest is the global organization of the strata of those systems for which the normalizing transformations converge, or for which they have integrable or linearizable saddles as $\lambda$ and the other parameters of the system vary. Many of their results are valid in the larger context of polynomial or analytic fields. They explain several features of the bifurcation diagram, namely, the existence of a continuous skeleton of integrable (linearizable) systems with sequences of holes filled with orbitally normalizable (normalizable) systems and strata finishing at a particular value of $\lambda ·$ In particular, they introduce the Ecalle-Voronin invariants of analytic classification of a saddle point or the Martinet-Ramis invariants for a saddle-node and illustrate their role as organizing centers of the bifurcation diagram.

##### MSC:
 37G05 Normal forms 34C20 Transformation and reduction of ODE and systems, normal forms 34C05 Location of integral curves, singular points, limit cycles (ODE)