# 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
*ω*_{1} and *ω*_{2} 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 *k*^{2} = Δ^{2} + 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 - k*^{2}/4 < Δ <
π + k^{2}/4.

It agrees well with the bifurcation picture above.

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*updated* 14 Nov 2006