Classifications of global attractors for S1-equivariant parabolic equations: A survey

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).