Control of evolutionary processes, topological index and deformation theory

Abstract

A general approach in studies of evolutionary processes is discussed. Such process can be described by a vector field with polynomial, analytic or smooth coefficients in phase space. The corresponding complex systems are investigated by perturbation analysis of the control and behavioral spaces together with associated bifurcation sets and discriminants. The approach is based essentially on the modern theory of deformations, the theory of logarithmic differential forms and integrable connections associated with deformations. Such a connection can be represented as a holonomic system of differential equations of Fuchsian type whose coefficients have logarithmic poles along the bifurcation set or discriminant of a deformation. The main tool in analysis of such objects is a new method for computing the topological index of a vector field.