Like the Mandelbrot fractals, the circle map has a basin of attraction
that can be viewed. This page contains movies that illustrate that
basin of attraction.
The following are the same as the
YouTube
page, but with different formats/resolutions.
All of the above movies show the basin of attraction of
z(n+1) = z(n) + omega - K * sin (2 * pi * z(n))
with z, omega and K complex, and Re z(n) == Re z(n) mod 1 (i.e.
we've wrapped the real part of z).
Re omega
Real part of omega -- Time axis -- i.e. Re omega = 0.0 at start of movie,
and Re omega = 1.0 at end of movie.
Im omega
Imaginary part of omega -- zero for the entire movie.
Re K
plotted along horizontal axis --
runs from -0.5 on left side of frame, to +0.5 on right side of
frame.
Im K
plotted along vertical axis -- is zero in the center. Aspect ratio
is preserved -- i.e it runs from -0.5* (240/320) to +0.5 * (240/320)
Note that the movies are essentially symmetric about omega=0.5.
The four movies above differ only in length, choice of colormap, and
length of iteration (the long movies were iterated more, and thus
have better resolution).
To get a better view of the overall,
click here
to see a short movie (185 KBytes),
with ReK running from -2.0 to 2.0, left to right.
Note that the ImK=0 axis will intersect an infinite number of
Arnold's tongues for any value of omega, at just about any value
of K, including very large K. Note, hoever, the tongues get
quite small as K gets large, as is very clear in the above movie.
Click here for a short movie (1.23 MBytes) or
here for a long
move (5.25 MBytes) showing Im omega held constant at 0.1.
Note that ReK runs from -0.2 to 0.2, left to right.
Click here for a short movie (460K)
showing Im omega held constant at 0.5.
Note that ReK runs from -0.2 to 0.2, left to right.
This (684K) and
this (852K, different colormap)
show ReK along x-axis, (running -1 to +1), ReOmega
along vertical axis, and ImK as time, running from 0.0 to 1.0.
Note that there is no periodicity here. ImOmega was held at 0.
Note that the very first frame of teh movie has ImK equal to
0.004. Note that this web site has numerous pictures with
ImK equal to zero -- turned on thier side. Note how even a
slight disturbance (a very small ImK) has all but wiped out
the Arnold's tongues with non-zero winding number.
These movies show the interior measure of the circle map, with
the parameters and values as explained at the top of this page.
The following are the same as the
YouTube
page, but with different formats/resolutions.