Functional Analysis and Its Applications

, Volume 46, Issue 3, pp 173–190

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



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 systems heat equation Chazy equation Darboux-Halphen system 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.V. A. Steklov Mathematical InstituteMoscowRussia

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