Iterations of real quadratic functions
We can trace real maps xn+1 = f( xn )
dynamics on 2D the "iteration diagram" to the left below. Dependence
xn on n is plotted to the right.
Controls: Drag the blue curve to change C and
xo 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 xo .
To plot the iteration diagram we draw the vertical red line from
xo toward the blue curve
y = f(x) = x2 + c, where
y1 = f(xo ).
To get the second iteration we draw the red horizontal line to the green
y = x one, where x1 = y1 =
f(xo ). Then draw again the vertical line to the blue curve
to get y2 = f(x1 ) and so on.
Points fc: xo → x1 →
x2 → ... for some value c and
xo form the orbit of xo.
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 x1
→ x2 → x1 ...
The map f o2 has two attracting fixed points
x1, x2 (on the right image).
Such qualitative change in map dynamics is called bifurcation.
Drag the blue curve to watch the bifurcation.
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Next: Bifurcation diagram
updated 21 Dec 2013