  In non-Hermitian crystal systems under open boundary condition (OBC), it is
generally believed that the OBC modes with frequencies containing positive
imaginary parts, when excited by external driving, will experience exponential
growth in population, thereby leading to instability. However, our work
challenges this conventional understanding. In such a system, we find an
anomalous response that grows exponentially with the frequency aligned with
those of saddle points. The frequencies of these saddle points on the complex
plane are below the maximum imaginary part of OBC spectrum, but they can lie
within or beyond the OBC spectrum. We derive general formulas of
excitation-response relationships and find that this anomalous response can
occur because the excitation of OBC modes eventually evolve toward these saddle
points at long times. Only when the frequencies of all these saddle points are
below the real axis do the non-Hermitian crystal systems remain stable under
periodic excitation. Thus our results also provide new insights on the
stability criterion of non-Hermitian crystal systems.
