Realization of meander permutations by boundary value problems

We consider Neumann boundary value problems of the form u(xx) + f(x, u, u(x)) = 0 oil the unit interval 0 less than or equal to x less than or equal to 1 for a certain class of dissipative nonlinearities f. Associated to these problems we have ii) meanders in the phase space (u, u(x)) is an element of R-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 iii) meander permutations pi(f) is an element of S(n) obtained by ordering the intersection points first along the u-axis and then along the meander. The meander permutation pi(f) is the permutation defined by the braid of solutions in the space (x, u, u(x)). It was recently shown by Fiedler and Rocha that n, determines the global attractor of the dynamical system generated by the semilinear parabolic differential equation u(t) = u(xx) + f(x, u, u(x)), up to C-0 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 ale realizable by the above boundary value problems. (C) 1999 Academic Press.