Abstract

Based on a Morse-Smale structure we study planar global attractors of the scalar reaction-advection-diffusion equation u_t = u_xx + ƒ(x, u, u_x) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of ƒ, and hyperbolicity of equilibria. We call Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type.

The planar Sturm attractor consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits. The unique heteroclinic orbits between adjacent Morse levels define a plane graph which we call the connection graph. Its 1-skeleton consists of the unstable manifolds (separatrices) of the index-1 Morse saddles.

We present two results which completely characterize the connection graphs and their 1-skeletons , in purely graph theoretical terms. Connection graphs are characterized by the existence of pairs of Hamiltonian paths with certain chiral restrictions on face passages. Their 1-skeletons are characterized by the existence of cycle-free orientations with only one maximum and only one minimum. Such orientations are called bipolar in [de Fraysseix, de Mendez, Rosenstiehl, Discr. Appl. Math. 56: 157–179, 1995].

In the present paper we show the equivalence of the two characterizations. Moreover we show that connection graphs of Sturm attractors indeed satisfy the required properties. In [Fiedler and Rocha, J. Diff. Equ. 244: 1255–1286, 2008] we show, conversely, how to design a planar Sturm attractor with prescribed plane connection graph or 1-skeleton of the required properties. In [Fiedler and Rocha, Connectivity and design of planar global attractors of Sturm type, III: Small and Platonic examples, 2008] we describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1-skeletons.