Polynomial dynamical systems and ordinary differential equations associated with the heat equation. (English. Russian original) Zbl 1275.35012
Funct. Anal. Appl. 46, No. 3, 173-190 (2012); translation from Funkts. Anal. Prilozh. 46, No. 3, 16-37 (2012).
Summary: 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.
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