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Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals

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Abstract

We study the bifurcations of two parameter families of circle maps that are similar tof b,w (x)=x+w+(b/2π) sin (2πx) (mod1). The bifurcation diagram is constructed in terms of setsT r , whereT r is the set of parameter values (b, w) for whichf b, w has an orbit with rotation numberr. We show that the known structure whenb<1 (forr rational,T r is an Arnol'd tongue and forr irrational, it is an arc) extends nicely into the regionb>1, wheref b, w is no longer injective and can have an interval of rotation numbers. Specifically, the tongues overlap in a uniform, monotonic manner and forr irrational,T r opens into a tongue. Our other theorems give information about the dynamics off b, w (e.g., bistability or aperiodicity) for (b, w) in various subsets of parameter space.

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  1. Philip L. Boyland

    Present address: Department of Mathematics, Maharishi International University, 52556, Fairfield, IA, USA

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  1. Department of Mathematics, Boston University, 02115, Boston, MA, USA

    Philip L. Boyland

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  1. Philip L. Boyland
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Communicated by O. E. Lanford

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Boyland, P.L. Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals. Commun.Math. Phys. 106, 353–381 (1986). https://doi.org/10.1007/BF01207252

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  • DOI: https://doi.org/10.1007/BF01207252

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