Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$u_t = u_{xx} + f(x,u,u_x)\,,$$ on the unit interval $0 < x<1$ with Neumann boundary conditions. Equilibria $v_t=0$ are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors $\mathcal{A}$ as 3-cell templates $\mathcal{C}$. The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries $x=0$ and $x=1$, respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors $\mathcal{A}$ with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors.Comment: 73+(ii) pages, 40 figures, 14 table; see also parts 1 and 2 under arxiv:1611.02003 and arxiv:1704.0034

in memoriam Klaus Kirchgässner

We study the boundary of unstable manifolds in parabolic partial differential equations of Sturm type. We show that the boundary naturally projects to a Schoenflies sphere. In particular this excludes complications like lens spaces, Reidemeister torsion, and nonmanifold boundaries.version 2 of October 11, 201

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

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{$*$} \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 $v_1 \leadsto v_2$ between equilibrium vertices $v_1, v_2$ 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. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graphs then also exhibit global time reversibility.Comment: 39+ii pages, 7 figure

Design of Sturm global attractors 1: Meanders with three noses, and reversibility

We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) u(t) = u(xx) + f(x, u, u(x)) 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 ordinary differential equation boundary value problem of equilibrium solutions u(t) = 0. 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, with cubic nonlinearity f = f(u), features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits v(1) (sic) v(2) between equilibrium vertices v(1), v(2) of adjacent Morse index. The global attractor turns out to be a ball of dimension d, given as the closure of the unstable manifold W-u(O) of the unique equilibrium with maximal Morse index d. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the (d - 1)-sphere boundary of the global attractor

Boundary Orders and Geometry of the Signed Thom–Smale Complex for Sturm Global Attractors

We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE ut=uxx+f(x,u,ux) (*) on the unit interval 0<x<1 with Neumann boundary conditions. We assume f to be dissipative with N hyperbolic equilibria v∈E. The global attractor A of (*), also called Sturm global attractor, consists of the unstable manifolds of all equilibria v. As cells, these form the Thom–Smale complex C. Based on the fast unstable manifolds of v, we introduce a refinement Cs of the regular cell complex C, which we call the signed Thom–Smale complex. Given the signed cell complex Cs and its underlying partial order, only, we derive the two total boundary orders hι:{1,…,N}→E of the equilibrium values v(x) at the two Neumann boundaries ι=x=0,1 . In previous work we have already established how the resulting Sturm permutation σ:=h−10∘h1, conversely, determines the global attractor A uniquely, up to topological conjugacy

A permutation characterization of Sturm attractors of Hamiltonian type

We consider Neumann boundary value problems of the form $u_t = u_{xx} + f$ on the interval $0 \leq x \leq \pi$ for dissipative nonlinearities $f = f (u)$. A permutation characterization for the global attractors of the semiflows generated by these equations is well known, even in the general case $f = f (x, u, u_x )$. We present a permutation characterization for the global attractors in the restrictive class of nonlinearities $f = f (u)$. In this class the stationary solutions of the parabolic equation satisfy the second order ODE $v^{\prime\prime} + f (v) = 0$ and we obtain the permutation characterization from a characterization of the set of $2\pi$-periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation

Heteroclinic orbits between rotating waves of semilinear parabolic equation on the circle

We investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction-advection-diffusion equations with periodic boundary conditions. Using zero number properties of the solutions and the phase shift equivariance of the equation, we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves