We consider homogeneous polynomial dynamical systems in the n-dimensional space, n = 0, 1, 2, …. For any such system the construction matches a non-linear ordinary differential equation. We describe the algorithm that brings the solution of such an equation to a solution of the heat equation. The classical fundamental solution of the heat equation corresponds to the case n = 0 in terms of our construction.

An explicit description of the family of ordinary differential equations arising in our methods will be given. This family contains equations with the Panlev e property, which arise in modern physical problems and the theory of integrable systems.

The course is based on resuls obtained recently with E. Yu. Bunkova. It is designed for a broad audience.