  Quantum chaos---the study of quantized nonintegrable Hamiltonian systems---is
an extremely well-developed and sophisticated field. By contrast, very little
work has been done in looking at quantum versions of systems which classically
exhibit {\it dissipative} chaos. Using the decoherence formalism of Gell-Mann
and Hartle, I find a quantum mechanical analog of one such system, the forced
damped Duffing oscillator. I demonstrate the classical limit of the system, and
discuss its decoherent histories. I show that using decoherent histories, one
can define not only the quantum map of an entire density operator, but can find
an analog to the Poincar\'e map of the individual trajectory. Finally, I argue
the usefulness of this model as an example of quantum dissipative chaos, as
well as of a practical application of the decoherence formalism to an
interesting problem.
