, Volume 46, Issue 3, pp 173-190
Date: 14 Sep 2012

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

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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.

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 46, No. 3, pp. 16–37, 2012
Original Russian Text Copyright © by V. M. Buchstaber and E. Yu. Bunkova
The work was supported by RFBR grants nos. 11-01-00197-a and 11-01-12067-ofi-m-2011 and by RF Government grant no. 2010-220-01-077, contract no. 11.G34.31.0005.