The Lorenz-Fermi-Pasta-Ulam experiment

We consider a chain of Lorenz '63 systems connected through a local, nearest-neighbour coupling. We refer to the resulting system as the Lorenz-Fermi-Pasta-Ulam lattice because of its similarity to the celebrated experiment conducted by Fermi, Pasta and Ulam. At large coupling strengths, the systems synchronize to a global, chaotic orbit of the Lorenz attractor. For smaller coupling, the synchronized state loses stability. Instead, steady, spatially structured equilibrium states are observed. These steady states are related to the heteroclinic orbits of the system describing stationary solutions to the partial differential equation that emerges on taking the continuum limit of the lattice. Notably, these orbits connect saddle-foci, suggesting the existence of a multitude of such equilibria in relatively wide systems. On lowering the coupling strength yet further, the steady states lose stability in what appear to be always subcritical Hopf bifurcations. This can lead to a variety of time-dependent states with fixed time-averaged spatial structure. Such solutions can be limit cycles, tori or possibly chaotic attractors. "Cluster states" can also occur (though with less regularity), consisting of lattices in which the elements are partitioned into families of synchronized subsystems. Ultimately, for very weak coupling, the lattice loses its time-averaged spatial structure. At this stage, the properties of the lattice are probably chaotic and approximately scale with the lattice size, suggesting that the system is essentially an ensemble of elements that evolve largely independent of one another. The weak interaction, however, is sufficient to induce widespread coherent phases; these are ephemeral states in which the dynamics of one or more subsystems takes a more regular form. We present measures of the complexity of these incoherent lattices, and discuss the concept of a "dynamical horizon" (that is, the distance along the lattice that one subsystem can effectively influence another) and error propagation (how the introduction of a disturbance in one subsystem becomes spread throughout the lattice). (C) 2000 Elsevier Science B.V. All rights reserved.