# Bifurcation diagram for quadratic maps

There is a good way to trace bifurcations on the *(x, c)*.
Let us plot iterations *f*_{c}: x_{o} = 0 →
x_{1} → x_{2} →...→ x_{MaxIt}
for all real *c*. Colors (from blue to red)
show how often an orbit visits the pixel.
You can watch iterations of *f*_{c}(x) for corresponding
*c* values on the right.
*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 = 2000.
See coordinates of the image center and *Δx, Δc* in the
text field. The vertical line goes through *x = 0*.
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 +∞ (see tangent bifurcation).
Filaments and broadening show how the critical orbit points are attracted to
the fixed point. Near *c = -3/4* we see a *branching point* due to
period doubling bifurcation. Then all the
Feigenbaum's bifurcation cascade.
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 the period-3 window.
Compare the map with the Mandelbrot set to the right.
## 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
*f*_{c}^{ on}(0).
Let us denote *g*_{n}(c) = f_{c}^{on}(0), then

*g*_{o}(c) = 0, g_{1}(c) = c,
g_{2}(c) = c^{2} + c, ...

The curves *g*_{0,1,...,6}(c) are shown in Fig.2.

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The M and J sets

*updated* 8 Sep 2013