We will consider homogeneous polynomial dynamical systems in an n-dimensional space, n = 0, 1, 2, 3, ... For any such system our construction matches a non-linear ordinary differential equation. We will describe the algorithm that brings the solution of such equation to a solution of the one-dimensional heat equation. The classical solution of the heat equation corresponds to the case n = 0. We will give the full classification of non-linear ordinary differential equation that arise from our construction. We will discuss in details the dynamical systems and differential equations which arise in classical problems of mechanics, in modern physical problems and in the theory of integrable systems. The lecture will be accessible to a broad audience.