More complicated analytic quadratic map is
xn+1 = fc( xn ) =
xn2 + c.
It appears e.g. in dynamics of biological populations. On complex plane it
generates the famous Mandelbrot and Julia fractal sets. In spite of apparent
simplicity it has very rich dynamics. For this map regions of regular and
chaotic dynamics are entangled in an intricate manner and scenarios of
transition to chaos are common for many other dynamical systems.
Dynamics of 1D real maps is useful to trace on iteration diagram
shown below. The blue curve is the N-th iteration
f oN(x) = f(f(...f(x))). Diagonal y = x is the green
line. -2 ≤ x,y ≤ 2. As since f(0) = C then for N = 1
the C value coincides with y(0).
Dependence xn on n is plotted in the right window.
Controls: Drag the blue curve by mouse to change C value.
Hold <Shift> to change the starting point xo.
Press <Enter> to set new parameters from the text fields.
To plot the first iteration we draw vertical red line from the
starting point xo = 0 toward the blue curve y = f(x) =
x2 + c, where yo = f(xo).
To get the second iteration we draw red horizontal line to the green
diagonal y = x, where x1 = yo =
f(xo). Then draw again vertical line to the blue curve to get
y1 = f(x1) and so on.
Points fc: xo → x1 →
x2 → ... for some c and xo values
make orbit of the point xo (it is plotted in the right
part of this applet).
For an analytic map points where f '(xc ) = 0 are called
critical points. Every stable cycle attracts at least one critical
point. Quadratic map has the only critical point xc = 0.
Therefore it can have only one attracting cycle and xc is
used usually as the starting point to find the cycle.
For C = -1/2 iterations go quickly 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 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 first derivative of a map at a fixed point
m = f '(x•) = 2x•
is called multiplier (or the eigenvalue) of the point.
For small enough δx
f(x• + δx) =
f(x•) + mδx +
O(δx2) ≈ x• + mδx.
So a fixed point is stable (attracting),
superstable, repelling, indifferent (neutral)
according as its multiplier satisfies |m| < 1,
|m| = 0, |m| > 1 or |m| = 1.
The second fixed point (the right intersection) is always repelling. For
|x| > x2 iterations go to infinity. For
|x| < x2 they go to the attracting fixed point
x1. This interval is the basin of attraction of
To the left on complex plane are shown: fractal basin of attractor
z1 (the blue region), repeller z2 and
two points of unstable complex cycle z3 , z4 .
For C = -1 the map has attractin period-2 cycle (the left picture
above). The second iteration of the map f o2
has two attracting fixed points z3 and z4 .
The basin of the cycle is shown to the left. f o2
attract the red regions to z3 and the blue ones to
z4 . z1 becomes a repeller.
For a map xn+1 = f(xn )
a small deviation δxo of coordinate
xo leads to a small change in x1
f '(xo) δxo.
For n iterations
δxo∏i=0,n-1 f '(xi ).
Then the Lyapunov exponent is determined as
Λ = limn → ∞ Ln ,
Ln = 1/n
log|δxn /δxo| =
1/n ∑i=0,n-1 ln |f '(xi )|.
For a chaotic orbit |δxn| grows with
increasing of n so Λ > 0.
You see below chaotic quadratic map for c = -2 with positive
Lyapunov exponent L calculated for shown finite orbit segment.
For attracting cycle below L is negative
Previous: Strange Cantor repeller
Next: Bifurcation diagram
updated 3 July 07