Double standard maps

We investigate the family of double standard maps of the circle onto itself, given by fa,b( x) = 2x + a + (b/pi) sin(2 pi x) (mod 1), where the parameters a, b are real and 0 <= b <= 1. Similarly to the well known family of (Arnold) standard maps of the circle, Aa, b(x) = x + a + (b/( 2 pi)) sin(2 pi x) ( mod 1), any such map has at most one attracting periodic orbit and the set of parameters (a, b) for which such orbit exists is divided into tongues. However, unlike the classical Arnold tongues that begin at the level b = 0, for double standard maps the tongues begin at higher levels, depending on the tongue. Moreover, the order of the tongues is different. For the standard maps it is governed by the continued fraction expansions of rational numbers; for the double standard maps it is governed by their binary expansions. We investigate closer two families of tongues with different behavior.