Iterations of real function xn+1 = f( xn ) = xn2 + c

We begin with this demonstration, where map f oN(x) = f(f(...f(x))) is the blue curve, y = x is the green line and -2 < x,y < 2. C axis coincides with the Y one because y(0) = f(0) = C. Dependence xn on n is ploted in the right window.

Drag mouse to change C value. Press <Enter> to set new parameters from the text fields.

The Mandelbrot set and Iterations

For more "words" and detailed explanations on functions, iterations and bifurcations for beginners look at "A closer look at chaos" and "Fractal Geometry of the Mandelbrot Set: I. The Periods of the Bulbs" by Robert L. Devaney.

The Mandelbrot set is built by iterations of function (map)
    zm+1 = f( zm ) = zm2 + c   or
    fc:  zo -> z1 -> z2 -> ...
for complex z and c. Iterations begin from starting point zo (usually zo = 0 + 0 i).
For real c and zo , zm are real too and we can trace iterations on 2D (x,y) plane. To plot the first iteration we draw vertical red line from xo = 0 toward blue curve y = f(x) = x2 + c, where y1 = f(xo) = c.

drag mouse to change the C value

To get the second iteration we draw red horizontal line to the green y = x line, where x1 = y1 = f(xo). Then draw again vertical line to the blue curve to get y2 = f(x1) and so on. Dependence of xm on m is plotted in the right part of this applet. Points fc: xo -> x1 -> x2 -> ... at some value c and xo form an orbit of   xo.
After 25 steps iterations go near to an attracting fixed point   x* = f(x*) of the map f. f doesn't move the point. Fixed points correspond to intersections of y = x and y = f(x) (green and blue) curves. There are always two fixed points for a quadratic map f because of two roots of quadratic equation
    f(x*) - x* = x*2 + c - x* = 0,
    x1,2 = 1/2 -+ (1/4 - c)1/2

The second fixed point (the right intersection) is repelling.
The roots may be complex for some c values. You can see here that for c > 1/4 attracting fixed point becomes repelling (and even complex therefore we can not see intersections on real plane). Iterations go to infinity. It is proven that an orbit go to infinity if |zn| > 2. Such qualitative change in iteration dynamics is called bifurcation.
Points on complex plane c that starting from zo = 0 don't go to infinity under iterations form the Mandelbrot set.
For c = -0.75 attracting fixed point becomes repelling and iterations converge to attracting period-2 orbit   x1 -> x2 -> x1 ... (see Birth of attracting period 2 orbit later).


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updated 5 February 2000