Differentiability at the Tip of Arnold Tongues for Diophantine Rotations: Numerical Studies and Renormalization Group Explanations

We study numerically the regularity of Arnold tongues corresponding to Diophantine rotation numbers of circle maps at the edge of validity of KAM theorem. This serves as a good test for the numerical stability of two different algorithms. We find empirically that Arnold tongues are only finitely differentiable at the tip. We also find several scaling properties of the Sobolev norms of the conjugacy near the breakdown. We also provide a renormalization group explanation of the regularity at the tip and the scaling behaviors of the Sobolev regularity. We also uncover empirically some other patterns which require explanation.