Circle Map Movies

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.

Short Blue (860 KBytes -- 60 frames)

Short Color (1.60 MBytes -- 60 frames)

Long Blue (4.80 MBytes -- 240 frames)

Long Color (7.45 MBytes -- 240 frames)

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.