This sequel continues our exploration arxiv:2302.12531 of a deceptively
``simple'' class of global attractors, called Sturm due to nodal properties.
They arise for the semilinear scalar parabolic PDE
\begin{equation}\label{eq:*}
u_t = u_{xx} + f(x,u,u_x) \tag{β}
\end{equation} on the unit interval 0<x<1, under Neumann boundary
conditions. This models the interplay of reaction, advection, and diffusion.
Our classification is based on the Sturm meanders, which arise from a
shooting approach to the ODE boundary value problem of equilibrium solutions
u=v(x). Specifically, we address meanders with only three ``noses'', each of
which is innermost to a nested family of upper or lower meander arcs. The
Chafee-Infante paradigm of 1974, with cubic nonlinearity f=f(u), features
just two noses.
We present, and fully prove, a precise description of global PDE connection
graphs, graded by Morse index, for such gradient-like Morse-Smale systems
\eqref{eq:*}. The directed edges denote PDE heteroclinic orbits v1ββv2β between equilibrium vertices v1β,v2β of adjacent Morse index. The
connection graphs can be described as a lattice-like structure of
Chafee-Infante subgraphs. However, this simple description requires us to
adjoin a single ``equilibrium'' vertex, formally, at Morse level -1.
Surprisingly, for parabolic PDEs based on irreversible diffusion, the
connection graphs then also exhibit global time reversibility.Comment: 39+ii pages, 7 figure