Iterations of real quadratic functions

Iteration diagram

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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updated 21 Dec 2013