Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals
Abstract
We study the bifurcations of two parameter families of circle maps that are similar to f _{ b,w }( x)= x+ w+( b/2π) sin (2π x) (mod1). The bifurcation diagram is constructed in terms of sets T _{ r }, where T _{ r } is the set of parameter values ( b, w) for which f _{ b, w } has an orbit with rotation number r. We show that the known structure when b<1 (for r rational, T _{ r } is an Arnol'd tongue and for r irrational, it is an arc) extends nicely into the region b>1, where f _{ b, w } is no longer injective and can have an interval of rotation numbers. Specifically, the tongues overlap in a uniform, monotonic manner and for r irrational, T _{ r } opens into a tongue. Our other theorems give information about the dynamics of f _{ b, w } (e.g., bistability or aperiodicity) for ( b, w) in various subsets of parameter space.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 September 1986
 DOI:
 10.1007/BF01207252
 Bibcode:
 1986CMaPh.106..353B