A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot{y}(t)=g(y(t))$ on ${\mathbb{R}}^d$ and those of the parabolic equations $\dot{u}=\mathrm{\Delta }u+f(x,u,\mathrm{\nabla }u)$ on a bounded domain $\mathrm{\Omega }$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\ge 3$ and in the corresponding cases $\mathrm{\Omega }=(0,1)$, $\mathrm{\Omega }={\mathbb{T}}^1$ and dim($\mathrm{\Omega }$)$\ge 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations.