Manchester Geometry Seminar 2012/2013

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.