FRONTS AND INTERFACES IN BISTABLE EXTENDED MAPPINGS

We study the time evolution of the interfaces in one-dimensional bista ble extended dynamical systems with discrete time. The dynamics are go verned by the competition between a local piecewise affine bistable ma pping and any couplings given by the convolution with a function of bo unded variation. We prove the existence of travelling wave interfaces, namely fronts, and the uniqueness of the corresponding selected veloc ity and shape. This selected velocity is shown to be the propagating v elocity for any interface, to depend continuously on the couplings and to increase with the symmetry parameter of the local nonlinearity. We apply the results to several examples including discrete and continuo us couplings, and the dynamics of planar fronts in multidimensional co upled map lattices. We use our technique to study the existence of oth er kinds of fronts. Finally we consider a more general class of bistab le extended mappings for which the couplings are allowed to be nonline ar and the local map to be smooth.