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.