  We discuss the quantum--classical correspondence in a specific dissipative
chaotic system, Duffing oscillator. We quantize it on the basis of quantum
state diffusion (QSD) which is a certain formulation for open quantum systems
and an effective tool for analyzing complex problems numerically. We consider a
sensitivity to initial conditions, `` pseudo-Lyapunov exponent '', and
investigate it in detail, varying Planck constant effectively. We show that in
a dissipative system there exists a certain critical stage in which the
crossover from classical to quantum behavior occurs. Furthermore, we show that
an effect of dissipation suppresses the occurrence of chaos in the quantum
region, while it, combined with the periodic external force, plays a crucial
role in the chaotic behaviors of classical system.
