Extended Symbolic Dynamics in Bistable CML: Existence and Stability of Fronts

Abstract

We consider a diffusive Coupled Map Lattice (CML) for which the local map is piece-wise affine and has two stable fixed points. By introducing a spatio-temporal coding, we prove the one-to-one correspondence between the set of global orbits and the set of admissible codes. This relationship is applied to the study of the (uniform) fronts' dynamics. It is shown that, for any given velocity in $[-1,1]$, there is a parameter set for which the fronts with that velocity exist and their shape is unique. The dependence of the velocity of the fronts on the local map's discontinuity is proved to be a Devil's staircase. Moreover, the linear stability of the global orbits which do not reach the discontinuity follows directly from our simple map. For the fronts, this statement is improved and as a consequence, the velocity of all the propagating interfaces is computed for any parameter. The fronts are shown to be also nonlinearly stable under some restrictions on the parameters. Actually, these restrictions follow from the co-existence of uniform fronts and non-uniformly travelling fronts for strong coupling. Finally, these results are extended to some $C^{\infty}$ local maps.