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Bifurcations of circle maps: Arnol’d tongues, bistability and rotation intervals. (English) Zbl 0612.58032
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 § 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.
Reviewer: C.Mira

37G99 Local and nonlocal bifurcation theory for dynamical systems
57R35 Differentiable mappings in differential topology
70K50 Bifurcations and instability for nonlinear problems in mechanics
47H10 Fixed-point theorems
Full Text: DOI
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