a very large number (typically $10^6$) initial conditions ${\rm{\bf X}}_{0,0}$ spread uniformly in $(r,p_r)$ phase space. Next we randomly choose a nearby phase space point ${\bf Y}_{0,0}$ within a circle of radious $d_0$, centered about ${\bf X}_{0,0}$. We calculate the Lyapunov exponent, using formula (25), from the successive evaluation of the distances $d_j$ for each $j$ iteration of the mapping. This process is repeated for several nearby ${\bf Y}_{0,0}$ trajectories. When the Lyapunov exponent reaches saturation we average the resulting value over the set of initial conditions to find $\lambda$. If this asymptotic value is positive, the system is defined as chaotic. To put a stricter criterion on the degree of chaos, we choose a threshold value of the exponent $\lambda_c$ beyond which the system is in the regime of hard chaos. We set $\lambda_c$ arbitrarily to the value 1, but as a check we generated Poincaré phase portraits to confirm chaos by looking for featureless ([*i.e.,*]{} no invariant tori) phase portraits. In this way, by varying the parameters $(\epsilon,{\tilde\omega_c})$ in a continuous fashion over the whole plane, running the map repeatedly and obtaining the resulting $\lambda$’s, we obtained the ‘phase diagram’ for this system, including a distinct ‘phase boundary’ 