Normalizable, Integrable, and Linearizable Saddle Points for Complex Quadratic Systems in $$\mathbb{C}^2 $$

AbstractIn this paper, we consider complex differential systems in the neighborhood of a singular point with eigenvalues in the ratio 1 : −λ with λ ∈ $$\mathbb{R}^{ + *} $$ . We 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 our study, we specialize to quadratic systems and study the values of λ for which these notions are distinct. For this purpose we give several tools for demonstrating normalizability, integrability, and linearizability.Our main interest is the global organization of the strata of those systems for which the normalizing transformations converge, or for which we have integrable or linearizable saddles as λ and the other parameters of the system vary. Many of the results are valid in the larger context of polynomial or analytic vector fields. We 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 λ. In particular, we introduce the Ecalle-Voronin invariants of analytic classifcation of a saddle point or the Martinet-Ramis invariants for a saddle-node and illustrate their role as organizing centers of the bifurcation diagram.