Design of Sturm global attractors 2: Time-reversible Chafee–Infante lattices of 3-nose meanders

Abstract

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 ut = uxx + f (x, u, ux ) (∗) on the unit interval 0 < x < 1, under Neumann boundary conditions. Thismodels 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 lowermeander 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. The directed edges denote PDE heteroclinic orbits v1 ↝ v2 between equilibrium vertices v1, v2 of adjacent Morse index. The connection graphs can be described as a latticelike 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.