  We study nonlinear phenomena of bistability and chaos at a level of few
quanta. For this purpose we consider a single-mode dissipative oscillator with
strong Kerr nonlinearity with respect to dissipation rate driven by a
monochromatic force as well as by a train of Gaussian pulses. The quantum
effects and decoherence in oscillatory mode are investigated on the framework
of the purity of states and the Wigner functions calculated from the master
equation. We demonstrate the quantum chaotic regime by means of a comparison
between the contour plots of the Wigner functions and the strange attractors on
the classical Poincar\'e section. Considering bistability at low-limit of
quanta, we analyze what is the minimal level of excitation numbers at which the
bistable regime of the system is displayed? We also discuss the formation of
oscillatory chaotic regime by varying oscillatory excitation numbers at ranges
of few quanta. We demonstrate quantum-interference phenomena that are assisted
hysteresis-cycle behavior and quantum chaos for the oscillator driven by the
train of Gaussian pulses as well as we establish the border of
classical-quantum correspondence for chaotic regimes in the case of strong
nonlinearities.
