Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals
This paper studies the bifurcations of families of noninvertible circle maps defined by \(x+w+bp(x)\) (mod 1), where (b,w) are two real parameters, p(x) being a continuous periodic function of period 1, which verifies other hypotheses given in {\S} 3. When \(p(x)\equiv (2\pi)^{-1}\sin (2\pi x)\), a ''standard'' map is obtained. More particularly, the author considers the bifurcation curves in the parameter plane (b,w), which are the boundaries of the existence domains of periodic points associated with a rational rotation number. In this case, such domains, with \(b>1\), are the extension (with overlappings) of what he calls ''Arnol'd tongues'' (known before Arnol'd in the theory of nonlinear harmonic and subharmonic synchronization of oscillators), obtained when the map is a diffeomorphism \((| b| <1\) and not \(b<1\) as indicated in the text). The case corresponding to irrational rotation numbers is also considered. Theorems give information about the dynamics (bistability and aperiodicity) of the maps for (b,w) belonging to various regions of the parameter plane.
scientific article; zbMATH DE number 3990452
scientific article