# 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)

`buchstab@mi.ras.ru`

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.