We survey the global dynamics of semiflows generated by scalar semilinear parabolic equations which are SO(2) equivariant under spatial shifts of x is an element of S-1 = R/2 pi Z, i.e. u(t) = u(xx) + f(u, u(x)), x is an element of S-1. (0.1) For dissipative C-2 nonlinearities f, the semiflow (0.1) possesses a compact global attractor A = A(f)(p) which we call Sturm attractor. The Sturm attractor A(f)(p) decomposes as A(f)(p) = epsilon(f) U F-f(p) U R-f(p) U H-f(p). where H-f(p) denotes heteroclinic orbits between distinct elements of spatially homogeneous equilibria epsilon(f) rigidly rotating waves R-f(p) and, as their non-rotating counterparts, frozen waves F-f(p) We therefore represent A(f)(p) by its connection graph C-f(p), with vertices in epsilon(f) F-f(p), R-f(p) and edges H-f(p). Under mild hyperbolicity assumptions, the directed graphs C-f(p) are finite and transitive. For illustration, we enumerate all 21 connection graphs C-f(p) with up to seven vertices. The result uses a lap signature of period maps associated to integrable versions of the steady state ODE of (0.1). Our results are based on a comparison with associated C-2 dissipative Neumann PDEs on the half-interval x is an element of (0, pi). After a suitable homotopy from to spatially reversible nonlinearities g(u, -p) = g(u, p), the Neumann problem gains a hidden symmetry O(2) spatially. Moreover, the Neumann dynamics become gradient-like. This leads to isomorphic connection graphs C-f(p) congruent to C-g(N) / similar to. Here similar to collapses the two shifted Neumann copies of each frozen or rotating wave to a single vertex. As an example, we freeze and reconstruct the connection graph of the Vas tulip attractor, known from delay differential equations, in setting (0.1).
Rocha et al. (Thu,) studied this question.