Localized solutions in lattice systems and their bifurcations caused by spatial interactions

We demonstrate the scenario showing how the stable spatially localized solutions with nontrivial (periodic, quasiperiodic or chaotic) dynamics may appear in lattice dynamical systems. It is important to mention that bifurcations to such regimes occur when the strength of spatial interactions exceeds some threshold. In fact we first prove the persistence of stationary localized structures in a range of weak interactions and then from this result of the 'anti-integrable limit' type we make the next step to show the existence of bifurcations of these states to the stable spatially localized states with a nontrivial time dynamics. We also show how our approach can be applied to sludy bifurcations to nonstationary states with spatial structure of general type.