Geodesic
Polynomial Dynamical Systems and Ordinary Differential Equations Associated with the Heat Equation
Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 3, pp. 16-37.

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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.
Keywords: polynomial dynamical systems, heat equation, Chazy equation, Darboux–Halphen system. 
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E. Yu. Bunkova; V. M. Buchstaber. Polynomial Dynamical Systems and Ordinary Differential Equations Associated with the Heat Equation. Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 3, pp. 16-37. http://geodesic.mathdoc.fr/item/FAA_2012_46_3_a1/

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