STURM GLOBAL ATTRACTORS FOR S1-EQUIVARIANT PARABOLIC EQUATIONS

We consider a semilinear parabolic equation of the form u(t) = u(xx) + f(u, v(x)) defined on the circle x is an element of S-1 = R/2 pi Z. For a dissipative nonlinearity f this equation generates a dissipative semiflow in the appropriate function space, and the corresponding global attractor A(f) is called a Sturm attractor. If f = f(u, p) is even in p, then the semiflow possesses an embedded flow satisfying Neumann boundary conditions on the half-interval (0, pi). This is due to O(2) equivariance of the semiflow and, more specifically, due to reflection at the axis through x = 0, pi is an element of S-1. For general f = f(u, p), where only SO(2) equivariance prevails, we will nevertheless use the Sturm permutation sigma introduced for the characterization of Neumann flows to obtain a purely combinatorial characterization of the Sturm attractors A(f) on the circle. With this Sturm permutation sigma we then enumerate and describe the heteroclinic connections of all Morse-Smale attractors A(f) with m stationary solutions and q periodic orbits, up to n := m + 2q <= 9.