Coalescing Complex Planar Stationary Points

Abstract

Among all bifurcation behaviors of analytic parametric families of real planar vector fields, those that stand out most prominently are confluences of distinct stationary points. The qualitative change is so drastic that in some classes of families (e.g., fold-like bifurcations) the stationary points leave the real plane altogether and slip into the complex plane. Although they disappear from the real domain they continue to organize the dynamics, and studying complex planar vector fields becomes a necessity even for real bifurcations.Our main concern is to describe à la Martinet-Ramis the analytical classification of generic holomorphic families unfolding a saddle-node vector field, and to relate this classification both to the dynamics of individual members of the family and to analytic properties of the saddle-node. For instance the problem of the existence of an analytic center-manifold for the saddle-node is characterized in terms of persistence (as the parameter tends to the bifurcation value) of heteroclinic connections between stationary points.We emphasize the geometric aspect of the classification. Complex trajectories are connected real surfaces allowing for richer geometric constructions as compared to 1-dimensional real trajectories. The trajectories are split by a finite collection of open “fibred squid sectors,” attached by spirals to stationary points within their adherence. The sectors are carved in such a way that one can construct an analytic and bounded conjugacy between the vector field and its formal normal form. The invariants of classification are obtained as transition maps of overlapping such normalization charts. Since we can perform this sectorial normalization analytically in the parameter, by restricting its values to “cells” covering the parameter space minus the bifurcation value, the resulting finite collection of functional invariants is analytic on parameter cells and continuous on their adherence. In that sense it “unfolds” Martinet-Ramis invariant of the saddle-node.The inverse problem (or realization) is addressed in the case of a persistent heteroclinic connections and provides unique normal forms (universal family for the analytic classification). We particularly show that in general the invariant cannot depend holomorphically on the parameter over a full neighborhood of the bifurcation value.