We study the solution behavior of a damped and parametrically driven
nonlinear chain modeled by a discrete nonlinear Schrodinger 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. [S1063-651X(99)04202-6].
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