Numerical computation of the asymptotic size of the rotation domain for the Arnold family

We consider the Arnold Tongue of the Arnold family of circle maps associated to a fixed Diophantine rotation number θ. The corresponding maps of the family are analytically conjugate to a rigid rotation. This conjugation is defined on a (maximal) complex strip of the circle and, after a suitable scaling, the size of this strip is given by an analytic function of the perturbative parameter. The main purpose of this paper is to perform a numerical accurate computation of this function and of its Taylor expansion. This allows us to verify previous theoretical results. The rotation numbers we select are quadratic irrationals, mainly the Golden Mean. By introducing a nonstandard extrapolation process, specially suited for the problem, we compute all the quantities required (rotation numbers, Arnold Tongues, Fourier and Taylor coefficients) with high precision. MSC : 37E10; 37F50; 65B05 PACS : 02.60.-x; 02.30.Nw; 02.30.Mv