  A novel class of nonequilibrium phase-transitions at zero temperature is
found in chains of nonlinear oscillators.For two paradigmatic systems, the
Hamiltonian XY model and the discrete nonlinear Schr\"odinger equation, we find
that the application of boundary forces induces two synchronized phases,
separated by a non-trivial interfacial region where the kinetic temperature is
finite. Dynamics in such supercritical state displays anomalous chaotic
properties whereby some observables are non-extensive and transport is
superdiffusive. At finite temperatures, the transition is smoothed, but the
temperature profile is still non-monotonous.
