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Bifurcation diagram for quadratic map
There
is a good way to trace bifurcations of quadratic map on the
(x, c)
plane by the
bifurcation diagram (it is very
similar to the "logistic
bifurcation map"). Let us plot iterations
fc: x1 = 0
→ x2
→ x3 → ...→ 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).
Usually initial points are
omitted but transient process shows how
fast iterations converge to an attractor.
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 begins with tangent bifurcation at
c
= 1/4.
For
c > 1/4 points go away to infinity.
For
-3/4 < c < 1/4 there is single attracting fixed
point.
Filaments and broadening show how the critical orbit points are attracted to
the fixed point. For
c = 0 the
fixed point coincides with the critical
one (the blue vertical line) and becomes superstable.
At
c ~ -3/4 we see a
branching point due to period doubling
bifurcation. After that the fixed point loses its stability and attracting
period-2
cycle appears. At
c = -1 the cycle goes through the critical
point and becomes superstable and so on. Cascade of period
doubling
bifurcations ends at
c = -1.401155. Further you see region of chaotic
dynamics. It ends by crisis at
c
= -2. For
c < -2 there
is an invariant Cantor set of unstable orbits with zero measure and
iterations of the
critical point diverge.
In the chaotic band there are white narrow holes of
windows of periodic
dynamics. The lowest
and biggest one corresponds to period-3 window.
There are 3 junction regions in it and under iterations critical orbit
jumps
sequentially between these regions.
The bifurcation map patterns
|
Fig.1 shows that caustics in distribution
of points of chaotic orbits are
generated by an extremum of a map. Therefore singularities (painted in the
red) on the bifurcation
diagram appear at images of the critical point
fc on(0).
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.
|
Note that attractor of the map is located between
g1(c)
and
g2(c) curves. If a caustic crosses the vertical
line
x = 0 then
gk(c)
= fcok(0) = 0.
I.e. there is a superstable period-
k cycle. In the neighbourhood
of
this
c value there is corresponding window of regular dynamics.
Contents
Previous:
Quadratic map
Next:
Windows of regular dynamics
scaling
updated 12 July 2006
(C) A. Eliëns
2/9/2007
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