# Iterations of real quadratic functions

## Iteration diagram

We can trace real maps *x*_{n+1} = f( x_{n} )
dynamics on 2D the "iteration diagram" to the left below. Dependence
*x*_{n} on *n* is plotted to the right.

*Controls:* Drag the blue curve to change *C* and
*x*_{o} value. Press <Enter> to set new parameter
values from text fields.
Here the blue curve is the map *f*^{ oN}(x) = f(f(...f(x))).
*-2 ≤ x, y ≤ 2*. For *N = 1* we get *y(0) = f(0) = C* and
*C* value coincides with *Y* coordinate. Iterations begin
from the starting point *x*_{o }.

To plot the iteration diagram we draw the vertical red line from
*x*_{o} toward the blue curve
*y = f(x) = x*^{2} + c, where
*y*_{1} = f(x_{o }).
To get the second iteration we draw the red horizontal line to the green
*y = x* one, where *x*_{1} = y_{1} =
f(x_{o }). Then draw again the vertical line to the blue curve
to get *y*_{2} = f(x_{1 }) and so on.
Points *f*_{c}: x_{o} → x_{1} →
x_{2} → ... for some value *c* and
*x*_{o} form the *orbit of x*_{o}.

## Fixed points, attracting cycles and bifurcations

*Fixed points* of a map
*x*_{∗} = f(x_{∗ })
correspond to intersections of the *y = x* and
*y = f(x)* (green and blue) curves. E.g. for *C = -1/2*
iterations go away unstable fixed point (the right intersection)
and converge to attracting fixed point to the left.
For *c < -0.75* the left attracting fixed point becomes repelling and
iterations converge to *attracting period-2 orbit x*_{1}
→ x_{2} → x_{1} ...
The map *f*^{ o2} has two attracting fixed points
*x*_{1}, x_{2} (on the right image).
Such qualitative change in map dynamics is called *bifurcation*.
Drag the blue curve to watch the bifurcation.

Contents
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Next: Bifurcation diagram

*updated* 21 Dec 2013