is very complicated, and very sensitive to initial conditions. By recording the values at each successive period, we obtain a surface-of-section of the trajectory of the particle in phase space.

There are three independent parameters in the problem: ${\tilde\omega_c}$, $\epsilon$ and ${\tilde\hbar}$. However, for quantitative correspondences to be made later with the quantum results, as mentioned earlier, we keep the angular momentum $J=\ell{\tilde\hbar}$ fixed, which reduces the number of parameters to the first two. The transition to chaos is manifested in the parameter space spanned by $(\epsilon,{\tilde\omega_c})$ (see Fig. 1). All of our subsequent results refer to this space. We did investigate the effects of varying $J$ by varying ${\tilde\hbar}$ for fixed $\ell$, and the results are even quantitatively very similar.

The first (and most obvious) evidence of chaotic behavior is seen in the Poincaré surface of section in $(r,p_r)$. In Figures 2(a)–(d) we show the sections corresponding to $\epsilon$ values of 0.5, 1.5, 1.95 and 2.5, respectively, while ${\tilde\hbar}=0.01$, ${\tilde\omega_c}=2.0$ and $\ell=5$ are held fixed. (The reason for this particular choice has to do with the $(\epsilon,{\tilde\omega_c})$ ‘phase diagram’ for this system, which is explained in more detail shortly.) In the quasi-integrable regime (Figs. 2(a),(b)), the phase space is 