This sequel continues our exploration (Fiedler and Rocha in Chaos 33:083127, 2023. https://doi.org/10.1063/5.0147634) of a deceptively “simple” class of global attractors, called Sturm due to nodal properties. They arise for the semilinear scalar parabolic PDE on the unit interval $$0< x<1$$
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 $$u=v(x)$$
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 $$f=f(u)$$
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. The directed edges denote PDE heteroclinic orbits "Equation missing" between equilibrium vertices $$v_1, v_2$$
v
1
,
v
2
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 $$-1$$
-
1
. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graphs then also exhibit global time reversibility.