Quasi-periodic dynamics and the circle maps

In multidimentional dynamical systems with continuous time oscillations with several different frequencies are possible. These frequencies depend on system parameters and if their ratio is an irrational number then quasi-periodic dynamics appear. Such quasi-periodic orbit with two frequencies ω₁ and ω₂ is situated on a torus. Crosscection of this torus by plane makes a closed curve. One can parametrize points of this curve by angular coordinate θ. An orbit starting at a point θ_n returns again to the plane in a point θ_n+1. In a common case this Poincare map is θ_n+1 = θ_n + f(θ_n).

Circle maps

If we keep in f(θ) only the first harmonic, then we get nonlinear circle maps
θ_n+1 = θ_n + Δ + k sin θ_n (mod 2π).
Its orbits are defined by winding number
w = lim_{n→ ∞} (θ_n - θ_o)/2πn
note that θ is not truncated to interval (0,2π) here.
For k = 0 the map is a uniform rotation of the circle by angle Δ, winding number is w = Δ/2π and there are no fixed points for Δ ≠ 0 . Rational values w = Δ/2π = p/q with integer p and q correspond to periodic orbits with "frequency" (p, q). Such orbits make p revolutions around the circle by q iterations. Irrational Δ/2π correspond to quasi-periodic orbits. Such orbit is dense on the circle.
For k < 1 circle maps are monotonic and invertible functions. There are periodic and quasi-periodic orbits with zero Lyapunov exponents in this region. For k > 1 maps are non-monotonic and non-invertible. There are chaotic orbits with positive Lyapunov exponents in this region.

Bifurcation (isoperiodic) diagram on the (Δ/2π, k/2π) plane. Click mouse in window to find period p of the point. Click mouse + <Alt>(<Ctrl>) to Zoom In(Out) 2 times.

Descending to k = 0 and θ/2π = p/q Arnold's tongues correspond to resonant periodic orbits (p, q). For k < 1 black regions correspond to quasi-periodic dynamics. Measure of these regions decreases to zero for k → 1.
For k > 1 in black chaotic see there are windows of regular dynamics (Milnor's swallow) again.

Periodic orbits

You can see in Fig.1 that for k > 0 the map may have attracting and repelling fixed points θ_o. From
θ_o = θ_o + Δ + k sin θ_o
it follows
-sin θ_o = Δ / k ≤ 1.
Thus for k ≥ Δ there is attracting 0/1 orbit (it corresponds to the first intersection of the blue curve with diagonal). For larger k period doubling cascade takes place. Attracting 0/1 orbit loses its stability when derivative of the map
1 + k cos θ_o = -1
at hyperbola k² = Δ² + 4.
The stable 1/1 orbit (it corresponds to the second intersection of the red curve with diagonal) is determined by
θ_o + 2π = θ_o + Δ + k sin θ_o.
It exists in the region k ≥ 2π - Δ. One can expand the second iteration of the circle map for small k and get for the 1/2 orbit
π - k²/4 < Δ < π + k²/4.
It agrees well with the bifurcation picture above.

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updated 14 Nov 2006