dominated by invariant tori, which are close to those of the unperturbed problem. As the value of ${\tilde\omega_c}$ is increased, the tori begin to break up, and isolated chaotic islands begin to appear (Fig. 2(c)), until finally, all evidence of invariant curves disappears and all we see is the uniform chaotic sea (Fig. 2(d)). These values of $(\epsilon,{\tilde\omega_c})$ corresponding to the integrable, intermediate and chaotic regions will be retained throughout what follows to make comparisons between the classical and quantum results.

Corresponding to the transition from regular to chaotic behavior, we begin to see the appearance of diffusive growth in the averaged energy (or squared momentum) of a localized ensemble of initial conditions. Figure 3(a) shows the average energy as a function of time for the parameters corresponding to the quasi-integrable regime, while Figure 3(b) corresponds to parameter values in the chaotic regime. In contrast to the behavior in the quasi-integrable regime, where the energy $E$ is regular, oscillatory quasiperiodic functioning of time around a constant value, in the chaotic regime $E$ grows linearly (or $p_r$ grows quadratically) with time. (Here and subsequently, ‘time’ refers to stroboscopic time, just after every kick ).

A quantitative measure of the degree of 