Parametric rigidity of the Hopf bifurcation up to analytic conjugacy

In this paper we prove that the time part of the germ of an analytic family of vector fields with a Hopf bifurcation is rigid in the parameter. Time parts are associated with the temporal invariant of the analytic classification. Because the eigenvalues at zero are complex conjugate, time parts usually unfold in the hyperbolic direction, where the singular points are linearizable. We first identify the time part of a generic conformal family and prove that any weak holomorphic conjugacy between two time parts yields a biholomorphism analytic in the parameter. The existence of Fatou coordinates in both the Siegel and in the Poincare domains plays a fundamental role in the proof of this result.