  We investigate the parametric evolution of riddled basins related to
synchronization of chaos in two coupled piecewise-linear Lorenz maps. Riddling
means that the basin of the synchronized attractor is shown to be riddled with
holes belonging to another basin in an arbitrarily fine scale, which has
serious consequences on the predictability of the final state for such a
coupled system. We found that there are wide parameter intervals for which two
piecewise-linear Lorenz maps exhibit riddled basins (globally or locally),
which indicates that there are riddled basins in coupled Lorenz equations, as
previously suggested by numerical experiments. The use of piecewise-linear maps
makes it possible to prove rigorously the mathematical requirements for the
existence of riddled basins.
