The Lorenz-Fermi-Pasta-Ulam experiment

We consider a chain of Lorenz '63 systems connected through a local, neares t-neighbour coupling. We refer to the resulting system as the Lorenz-Fermi- Pasta-Ulam lattice because of its similarity to the celebrated experiment c onducted by Fermi, Pasta and Ulam. At large coupling strengths, the systems synchronize to a global, chaotic orbit of the Lorenz attractor. For smalle r coupling, the synchronized state loses stability. Instead, steady, spatia lly structured equilibrium states are observed. These steady states are rel ated to the heteroclinic orbits of the system describing stationary solutio ns to the partial differential equation that emerges on taking the continuu m limit of the lattice. Notably, these orbits connect saddle-foci, suggesti ng the existence of a multitude of such equilibria in relatively wide syste ms. On lowering the coupling strength yet further, the steady states lose s tability in what appear to be always subcritical Hopf bifurcations. This ca n lead to a variety of time-dependent states with fixed time-averaged spati al 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 o f the lattice are probably chaotic and approximately scale with the lattice size, suggesting that the system is essentially an ensemble of elements th at evolve largely independent of one another. The weak interaction, however , is sufficient to induce widespread coherent phases; these are ephemeral s tates in which the dynamics of one or more subsystems takes a more regular form. We present measures of the complexity of these incoherent lattices, a nd discuss the concept of a "dynamical horizon" (that is, the distance alon g the lattice that one subsystem can effectively influence another) and err or propagation (how the introduction of a disturbance in one subsystem beco mes spread throughout the lattice). (C) 2000 Elsevier Science B.V. All righ ts reserved.