The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations y˙(t)=g(y(t)) on Rd and those
of the parabolic equations u˙=Δu+f(x,u,∇u) on a bounded
domain Ω. We give details on the similarities of these dynamics in the
cases d=1, d=2 and d≥3 and in the corresponding cases Ω=(0,1),
Ω=T1 and dim(Ω)≥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