We consider nonlinear dynamics in a finite parity-time-symmetric chain of the
discrete nonlinear Schrödinger (dNLS) type. We work in the range of the
gain and loss coefficient when the zero equilibrium state is neutrally stable.
We prove that the solutions of the dNLS equation do not blow up in a finite
time and the trajectories starting with small initial data remain bounded for
all times. Nevertheless, for arbitrary values of the gain and loss parameter,
there exist trajectories starting with large initial data that grow
exponentially fast for larger times with a rate that is rigorously identified.
Numerical computations illustrate these analytical results for dimers and
quadrimers.