On Parameter Space of Complex Polynomial Vector Fields in the Complex Plane

TL;DR

This paper investigates the geometric and topological structure of the parameter space of complex polynomial vector fields, revealing how bifurcations and stability of separatrices occur under perturbations.

Contribution

It provides a detailed analysis of the loci in parameter space with fixed topological structure and demonstrates the formation of homoclinic separatrices during bifurcations.

Findings

Homoclinic separatrices can form under small perturbations.

Decomposition of parameter space by combinatorial data is not a cell decomposition.

Landing separatrices are stable if equilibrium multiplicities are preserved.

Abstract

The space of degree d single-variable monic and centered complex polynomial vector fields can be decomposed into loci in which the vector fields have the same topological structure. We analyze the geometric structure of these loci and describe some bifurcations, in particular, it is proved that new homoclinic separatrices can form under small perturbation. By an example, we show that this decomposition of parameter space by combinatorial data is not a cell decomposition. The appendix to this article, joint work with Tan Lei, shows that landing separatrices are stable under small perturbation of the vector field if the multiplicities of the equilibrium points are preserved.