Scaling of the Arnold Tongues

Abstract:

When two oscillatorsare coupledtogetherthere 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 tonguesas a function of the periodq, winding numberp / 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, AQ, scales as kq, as originally suggested by Arnold. In the limit as q + m at fixed k , however, AS2 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 Amold tongues is a fat fractal, with an exponent that is 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.