  We show that the rate of increase of von Neumann entropy computed from the
reduced density matrix of an open quantum system is an excellent indicator of
the dynamical behavior of its classical hamiltonian counterpart. In decohering
quantum analogs of systems which exhibit classical hamiltonian chaos entropy
production rate quickly tends to a constant which is given by the sum of the
positive Lyapunov exponents, and falls off only as the system approaches
equilibrium. By contrast, integrable systems tend to have entropy production
rate which decreases as $t^{-1}$ well before equilibrium is attained. Thus,
behavior of quantum systems in contact with the environment can be used as a
test to determine the nature of their hamiltonian evolution.
