Calculus of variations is mostly preoccupied with minimizers or critical points, their existence, regularity, and sometimes their (non)uniqueness. For gradient flows

,

these just constitute time-independent equilibria . Connecting orbits , which run heteroclinically from a source equilibrium at to another target equilibrium at , are encoded in the Morse complex, but have much been neglected since then.

As a general class of examples, we study scalar parabolic PDEs in one space dimension with Neumann boundary conditions. Their global attractors consist precisely of the critical points and the connecting orbits of an associated gradient functional . We present elements of a combinatorial description of the resulting geometric Thom-Smale complexes. Already in three dimensions, surprising examples and obstructions arise. We also hope to present some recent results on a minimax characterization of boundary orders via nodal properties, on "discretizations" of minimal dimension, and on parabolic time reversal within global attractors.

This is joint work with Carlos Rocha. See also http://dynamics.mi.fu-berlin.de/