  We consider a Klein-Gordon chain that is periodically driven at one end and
has dissipation at one or both boundaries. An interesting numerical observation
in a recent study~[arXiv:2209.03977] was that for driving frequency in the
phonon band, there is a range of values of the driving amplitude $F_d\in (F_1,
F_2)$ over which the energy current remains constant. In this range, the system
exhibits a "resonant nonlinear wave" (RNW) mode of energy transmission which is
a time and space periodic solution. It was noted that the range $(F_1,F_2)$,
for which the RNW mode occurs, shrinks with increasing system size $N$ and
disappears eventually. Remarkably, we find that the RNW mode is in fact a
stable solution even for $F_d$ much larger than $F_2$ and quite large $N$
($\approx 1000$). For $F_d>F_{2}$, there exists a second attractor which is
chaotic. Both attractors have finite basins of attraction and can be reached by
appropriate choice of initial conditions. Corresponding to the two attractors
for large $F_d$, the system can now be in two nonequilibrium steady states. We
improve the perturbative treatment of [arXiv:2209.03977] for the RNW mode by
including the contributions of the third harmonics. We also consider the effect
of thermal noise at the boundaries and find that the RNW mode is stable for
small temperatures. Finally, we present results for a different driving
protocol studied in [arXiv:2205.03839] where $F_d$ is taken to scale with
system size as $N^{-1/2}$ and there is dissipation only at the non-driven end.
We find that the steady state can be characterized by Fourier's law as in
[arXiv:2205.03839] for a stochastic model. We point out interesting differences
that occur since our dynamics is nonlinear and Hamiltonian. Our results suggest
the intriguing possibility of observing the high current carrying RNW phase in
experiments by careful preparation of initial conditions.
