Periodic, quasiperiodic, and chaotic localized solutions of a driven, damped nonlinear lattice

We study the solution behavior of a damped and parametrically driven nonlinear chain modeled by a discrete nonlinear Schrödinger equation. Special attention is paid to the impact of the damping and driving terms on the existence and stability of localized solutions. Dependent upon the strength of the driving force, we find rich lattice dynamics such as stationary solitonlike solutions and periodic and quasiperiodic breathers, respectively. The latter are characterized by regular motion on tori in phase space. For a critical driving amplitude the torus is destroyed in the course of time, leaving temporarily a chaotic breather on the lattice. We call this order-chaos transition a dynamical quasiperiodic route to chaos. Eventually the chaotic breather collapses to a stable localized multisite state. Finally, it is demonstrated that above a certain amplitude of the parametric driving force no localized states exist.