Functional Analysis and Its Applications

, Volume 46, Issue 3, pp 173–190

Polynomial dynamical systems and ordinary differential equations associated with the heat equation


DOI: 10.1007/s10688-012-0024-2

Cite this article as:
Buchstaber, V.M. & Bunkova, E.Y. Funct Anal Its Appl (2012) 46: 173. doi:10.1007/s10688-012-0024-2


We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations.

Key words

polynomial dynamical systemsheat equationChazy equationDarboux-Halphen system

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.V. A. Steklov Mathematical InstituteMoscowRussia