Bifurcation diagram for quadratic maps
There is a good way to trace bifurcations of period of attracting orbit
on the (x, c) plane by the bifurcation diagram of f
(it is very similar to the "logistic
bifurcation map"). Let us plot iterations fc: xo
= 0 -> x1 -> x2 ->...-> xMaxIt for all real
c on the (x, c) plane. Colors (from blue to red) show how often
an orbit visits the pixel (colors are changed under zooming).
You can watch iterations of fc(x) for corresponding
c values in the right applet.
Controls: Click mouse to zoom in 2 times. Click mouse with
Ctrl to zoom out. Hold Shift key to zoom in the c
(vertical) direction only. Max number of iterations = 8000.
See coordinates of the image center and Dx,
Dc in the text field.
The vertical line goes through x = 0.
Compare the map with the rotated Mandelbrot set on the right.
The top part of the picture corresponds to a single attracting fixed point
of f for -3/4 < c < 1/4. For c > 1/4 points go away
to +Infinity (see tangent bifurcation).
Filaments and broadening show how the critical orbit points are attracted to
the fixed point. At c ~ -3/4 we see a branching point due to
period doubling bifurcation. Then all the
Feigenbaum's cascade of bifurcations.
At the lower part of the bifurcation diagram you see chaotic bands and
white narrow holes of windows of periodic dynamics. The lowest and
biggest one corresponds to period-3 window (there are 3 junction points in it).
The bifurcation map patterns
Fig.1 shows that caustics in distribution of points of chaotic orbits are
generated by an extremum of a mapping. Therefore singularities (painted in the
red) on the bifurcation diagram appear at images of the critical point
Let us denote gn(c) = fcon(0), then
go(c) = 0, g1(c) = c,
g2(c) = c2 + c, ...
The curves g0,1,...,6(c) are shown in Fig.2.
2D Real Mandelbrot + Julia set
To plot "2D Real Mandelbrot + Julia" set on the real
(xo , c) plane, you, starting from different
xo , repeat for each value of c transformation
xn+1 = xn2 + c (up to maximum number
of iterations), exiting if the magnitude of |xn| > 2.
If you finish the loop, the point is probably inside the Mandelbrot + Julia set
(the black region). If you exit, the point is outside and can be colored
according to how much iterations n were completed.
The Y axis corresponds to different c values and the
X axis corresponds to different starting points xo.
One can consider the set as a cross section of the Complex 4D Mandelbrot +
Julia set by the (zRe, cRe) plane too.
Bifurcation map (the right picture) compliments the "spider" and shows
Previous: Iterations of real maps
Next: Iterations on complex plane.
The M and J sets
updated 14 July 2002