On the classical-quantum correspondence for periodically time dependent systems

A simple, but instructive approach to quantum dynamics close to classical nonlinear resonances is discussed. It is shown that a quantum system can be fairly insensitive to a transition to chaos in the corresponding classical system; the same approximation can describe states on both sides of a ‘chaos border’. Tunneling between quantized invariant manifolds in periodically driven systems, as well as ‘scarring’ of Floquet states, is described analytically.