Manchester Geometry Seminar 2012/2013

Thursday 25 October 2012. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm

Algebraically Integrable Quadratic Dynamical Systems

Victor Buchstaber (Steklov Mathematical Institute and University of Manchester)

In the talk we present results obtained recently with E. Yu. Bunkova. Let us consider homogeneous quadratic dynamical systems in $n$-dimensional space. For such systems we introduce the notion of an algebraical integrability by a set of functions. We describe a wide class of quadratic dynamical systems that are algebraically integrable by a set of functions $h_1$, ..., $h_n$, where the first function $h_1$ is a solution of an ordinary differential equation of order $n$ and the functions $h_2$, ..., $h_n$ are differential polynomials in $h_1$. The class of algebraically integrable quadratic dynamical systems contains the classical three-dimensional Darboux-Halphen systems. We discuss modern generalizations of this systems, including new four-dimensional dynamical systems. We describe the class of ordinary differential equations on the basic function $h_1$ defining the set of integrating functions. In the case of the classical Darboux-Halphen system it is the famous Chazy equation. We show that our class of ordinary differential equations contains other equations with the Painleve property.

The talk is oriented toward a broad audience. Main definitions will be introduced during the talk.