Realization of Meander Permutations by Boundary Value Problems

We consider Neumann boundary value problems of the form uxx+f(x, u, ux)=0 on the unit interval 0 x 1 for a certain class of dissipative nonlinearities f. Associated to these problems we have (i) meanders in the phase space (u, ux) 2, which are connected oriented simple curves on the plane intersecting a fixed oriented line (the u-axis) in n points corresponding to the solutions; and (ii) meander permutations πf S(n) obtained by ordering the intersection points first along the u-axis and then along the meander. The meander permutation πf is the permutation defined by the braid of solutions in the space (x, u, ux). It was recently shown by Fiedler and Rocha that πf determines the global attractor of the dynamical system generated by the semilinear parabolic differential equation ut=uxx+f(x, u, ux), up to C0 orbit equivalence. Therefore, these permutations are of considerable importance in the classification problem of the (Morse–Smale) attractors for these dynamical systems. In this paper we present a purely combinatorial characterization of the set of meander permutations that are realizable by the above boundary value problems.