Connectivity and design of planar global attractors of Sturm type. II: Connection graphs

Based on a Morse–Smale structure we study planar global attractors of the scalar reaction–advection–diffusion equation ut=uxx+f(x,u,ux) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f, 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 certain restrictions on their criticality. Such orientations are called bipolar in [H. de Fraysseix, P.O. de Mendez, P. Rosenstiehl, Bipolar orientations revisited, Discrete Appl. Math. 56 (1995) 157–179]. In [B. Fiedler, C. Rocha, Connectivity and design of planar global attractors of Sturm type. I: Orientations and Hamiltonian paths, Crelle J. Reine Angew. Math. (2007), in press] we have shown the equivalence of the two characterizations. Moreover we have established that connection graphs of Sturm attractors indeed satisfy the required properties. In the present paper we show, conversely, how to design a planar Sturm attractor with prescribed plane connection graph or 1-skeleton of the required properties. In [B. Fiedler, C. Rocha, Connectivity and design of planar global attractors of Sturm type. III: Small and Platonic examples, 2007, submitted for publication] we describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1-skeletons.