On the full quantum description of nonlinear dynamic systems in the phase-space representation

An approach to the completely quantized description of nonlinear dynamic systems with chaos is considered. The method bases on the correspondence between operators and their symbols in a phase-space representation. Heisenberg equations in a wide class of phase-space representations are obtained. Nonlinear oscillations of the coupled quasiparticles-oscillator system are analyzed in the completely quantized description in terms of symbols of operators in the Wigner representation. The asymptotic solutions of the Heisenberg equations for the symbols Qt, Pt, etc. are obtained for a stable region of the phase-space, and the respective quantum observables are calculated.