SEMICLASSICAL QUANTIZATION OF KAM RESONANCES IN TIME-PERIODIC SYSTEMS

A semiclassical theory for the quasi-energy spectrum of time-periodic systems with accidental classical resonances is presented. The priMiti Ve EBK quantum conditions for integrable systems are extended to multi ply periodic flux tubes occuring in resonant systems. Replacing classi cal actions by appropriate differential operators in a classical reson ance Hamiltonian yields a uniform quantization of states related to a classical resonance region. The derivation being general for time-peri odic systems unfolds the organization of the quasi-energy spectrum, re ducing it to the spectrum of a single time-independent Hamiltonian of one degree of freedom with additional rational shifts of homegaBAR. In a first-order approximation the resonance Hamiltonian is reduced to a pendulum leading to a differential equation of the Mathieu type for t he quasi-energies. It is rigorously shown how parameters of the differ ential equation can be drawn from classical dynamics, using the data o f the 'essential' orbits in the resonance zone. i.e. stability coeffic ients and actions of hyperbolic and elliptic orbits as well as actions of homoclinic orbits. The quasi-energy spectrum of a forced quartic o scillator is studied numerically and evaluated. Semiclassical quasi-en ergies related to a resonance of period three are computed and compare d with exact quantum mechanical eigenvalues.