Double devil’s staircase in circle maps

AbstractAn orbit of the shift σ: t ↦ 2t on the circle = ℝ/ℤ is ordered if and only if it is contained in a semi-circle Cμ = [μ, μ+½]. We investigate the ‘devil's staircase’ associating to each μ ε the rotation number ν of the unique minimal closed σ-invariant set contained in Cμ; we present algorithms for μ in terms of ν, and we prove (after Douady) that if ν is irrational then μ is transcendental. We apply some of this analysis to questions concerning the square root map, and mode-locking for families of circle maps, we generalize our algorithms to orbits of the shift having ‘sequences of rotation numbers’, and we conclude with a characterization of all orders of points around realizable by orbits of σ.

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Bifurcation to Badly Ordered Orbits in One-Parameter Families of Circle Maps or Angels Fallen from the Devil's Staircase

Proceedings of the American Mathematical Society ◽

10.2307/2047355 ◽

1988 ◽

Vol 102 (4) ◽

pp. 1031 ◽

Cited By ~ 1

Author(s):

Kevin Hockett ◽

Philip Holmes

Keyword(s):

Devil's Staircase ◽

Circle Maps ◽

Devil’S Staircase

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Circle Maps and the Devil's Staircase in a Periodically Perturbed Oregonator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497001783 ◽

1997 ◽

Vol 07 (11) ◽

pp. 2621-2628 ◽

Cited By ~ 9

Author(s):

Morten Brøns ◽

Peter Gross ◽

Kedma Bar-Eli

Keyword(s):

Periodic Solutions ◽

Invariant Manifold ◽

Parameter Range ◽

Devil's Staircase ◽

Invariant Circle ◽

Small Oscillations ◽

Scaling Properties ◽

Circle Maps ◽

Devil’S Staircase ◽

Manifold Theory

Markman and Bar-Eli has studied a periodically forced Oregonator numerically and found a parameter range with the following properties: (1) Only periodic solutions are found in frequency-locked steps, each with a certain pattern of large and small oscillations (2) Between any two steps there is a step with the period being the sum of the two periods and the concatenation of the two patterns (3) Certain scaling properties as the period tends to infinity. We show that such behavior occurs if the dynamics of the system is governed by a family of diffeomorphisms of a circle with a Devil's staircase. Using invariant manifold theory we argue that an invariant circle must indeed exist when, as in the present case, the unforced system is close to a saddle-loop bifurcation. Generalizations of the results are briefly discussed.

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