Scaling of the Arnold tongues

When two oscillators are coupled together there are parameter regions called 'Arnold tongues' where they mode lock and their motion is periodic with a common frequency. We perform several numerical experiments on a circle map, studying the width of the Arnold tongues as a function of the period q, winding number p/q, and nonlinearity parameter k, in the subcritical region below the transition to chaos. There are several interesting scaling laws. In the limit as k -> 0 at fixed q, we find that the width of the tongues, Delta Omega, scales as k(q), as originally suggested by Arnold. In the limit as q -> infinity at fixed k, however, Delta Omega scales as q(-3), just as it does in the critical case. In addition, we find several interesting scaling laws under variations in p and k. The q(-3) scaling, token together with the observed p scaling, provides evidence that the ergodic region between the Arnold tongues is a fat fractal, with an exponent that is 2/3 throughout the subcritical range. This indirect evidence is supported by direct calculations of the fat-fractal exponent which yield values between 0.6 and 0.7 for 0.4 < k < 0.9.