Iterations of real quadratic functions
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.
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∗ =
x∗2 + c - x∗ = 0,
x1,2 = 1/2 ∓ (1/4 - c)½
The first derivative of quadratic map at a fixed point
λ = f '(z∗ ) =
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
Such qualitative change in map dynamics is called bifurcation.
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