We study oscillations of a spaser driven by an external optical wave. When the frequency of the external field is shifted
from the frequency of an autonomous spaser, the spaser exhibits stochastic oscillations at low field intensity. The
plasmon oscillations lock to the frequency of the external field only when the field amplitude exceeds a threshold value.
We find a region of external field amplitude and the frequency detuning (the Arnold tongue) for which the spaser
becomes synchronized with the external wave. We obtain the conditions upon the amplitude and frequency of the
external field (the curve of compensation) at which the spaser’s dipole moment oscillates with a phase shift of π
relatively to the external wave. For these values of the amplitude and frequency, the loss in the metal nanoparticles
within the spaser is exactly compensated for by the gain. It is expected that if these conditions are not satisfied, then due
to loss or gain of energy, the amplitude of the wave travelling along the system of spasers either tends to the curve of
compensation or leave the Arnold tongue. We also consider cooperative phenomena showing that in a chain of
interacting spasers, depending on the values of the coupling constants, either all spasers oscillate in phase or a nonlinear
autowave travels in the system. In the latter scenario, the traveling wave is harmonic, unlike excitations in other
nonlinear systems. Due to the nonlinear nature of the system, any initial distribution of spaser states evolves into one of
these steady states.