Scaling of the Arnold tongues

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Abstract
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. The authors 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 to 0 at fixed q, they find that the width of the tongues, Delta Omega , scales as kq, as originally suggested by Arnold (1961). In the limit as q to infinity at fixed k, however, Delta Omega scales as q-3, just as it does in the critical case. In addition, they find several interesting scaling laws under variations in p and k. The q-3 scaling, taken 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.