# Iterations of real quadratic functions

## Iteration diagram

We can trace real maps xn+1 = f( xn ) dynamics on 2D "iteration diagram" to the left below.
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. Iterations begin from the starting point xo . For N = 1 we get y(0) = f(0) = C and C value coincides with Y coordinate. Dependence xn on n is ploted on the right.
To plot iteration diagram we draw 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 red horizontal line to the green y = x one, where x1 = y1 = f(xo ). Then draw again vertical line to the blue curve to get y2 = f(x1 ) and so on.

Points fc: xo → x1 → x2 → ... at some value c and xo form an orbit of xo.

## Fixed points

For C = -1/2 iterations converge to an attracting fixed point   x = f(x) of the map. 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 = x2 + c - x = 0,     x1,2 = 1/2 ∓ (1/4 - c)½
The first derivative of quadratic map at a fixed point
λ = f '(z) = 2z
is a complex number called multiplier (or the eigenvalue) of the point. For small enough ε
f(z + ε) = f(z) + λ ε + O(ε2)
so a fixed point is either attracting or repelling or indifferent (neutral) according as its multiplier satisfies |λ| < 1 or |λ| > 1 or |λ| = 1.
The second fixed point (the right intersection) is always repelling. For c > 1/4 attracting fixed point becomes repelling (and even complex therefore we can not see intersections on real plane). Iterations diverge to infinity. Such qualitative change in map dynamics is called bifurcation.

## Attracting cycles

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). The map f o2 has two attracting fixed points x1, x2 (on the right image).
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updated 5 Nov 2013