Chaotic quadratic maps

Fig.1 illustrates stretching and folding transformations for the quadratic maps fc (for example the Myrberg-Feigenbaum point c = -1.401155 is chosen). The segment Ic = [-x2 , x2] is mapped into itself (here x2 = 1/2 + (1/4 - c)1/2 is the right repelling fixed point). Points outside Ic go to infinity. We see that after one application of fc , there are no points in [-x2 , c). The segment (c2+c, x2] is stretched every iteration. Points leave it and never return back. Thus eventually all points from Ic come into [c, c2+c] attractor, bounded by the g1(c) = c and g2(c) = c2+c curves.
Colors on the left image below show how fast iterations starting at a pixel go to infinity. "Whiskers" at the bottom correspond to the cantor repeller. Note that iterations always diverge for |x| > 2. You see below intervals Ic (the black region) and chaotic attractors (to the right) for different c values.

Chaotic dynamics for c = -2

Let us consider quadratic maps for c = -2
    xn+1 = xn2 - 2 .
It maps the interval [-2,2] onto itself. The map is contracting for |x| < 1 and expanding otherwise.
After substitution x = 2 cos(πy) where -2≤ x≤ 2 and 0≤ y≤1 we get
    cos(πyn+1) = 2 cos2(πyn ) - 1 = cos(2πyn ).
For yn ≤ 1/2 it follows
    yn+1 = 2yn .
For yn > 1/2 by means of formula cos(2πyn ) = cos(2π-2πyn ) we reduce cosine argument to the interval [0,π] and get
    yn+1 = 2 - 2yn .
It is chaotic tent map. Therefore quadratic map for c = -2 also has dense set of unstable periodic orbits and continuum of chaotic orbits.

Invariant densities

From x = 2 cos(πy) one gets
    |dx| = 2π |sin(πy)| dy = π(4 - x2)1/2 dy.
Chaotic tent map (as like as the sawtooth map) has the uniform density ρ(y)=1. Relative number of points of a chaotic orbit in a small interval dy is ρ(y)dy = dy. As since all these points are mapped in interval dx, therefore the number is equal to ρ(x)dx = dy and corresponding invariant density is
    ρ(x) = dy / |dx| = 1/π(4 - x2) -1/2.
The average expansion along a chaotic orbit for c = -2 is
    ∫ |2x| ρ(x) dx = 8/π = 2.546 .
The density is shown qualitatively to the left in blue-green-red colors (see bifurcation diagram).


Similar densities for band merging and interior crisis points are shown above.

You see below more complicated chaotic attractor and invariant density

Lyapunov exponent

For a map xn+1 = f(xn ) a small deviation δxo of coordinate xo leads to a small change in x1
    δx1 = f '(xo) δxo.
For n iterations
    δxn = 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.
As since for quadratic maps f '(x) = 2x therefore for c = -2 the Lyapunov exponent is
    λ = ∫ ln|2x| ρ(x) dx = ln 2 = 0.693 .
You see below chaotic quadratic map for c = -2. Drag xo (with <Shift>) to test that Lyapunov exponents L (calculated for shown finite orbit segments) are close to the precise value Λ and orbits are chaotic (i.e. L > 0) for almost all initial points (except a set of point with zero measure e.g. xo = 0).

Controls: drag the blue curve by mouse to change C value. Hold <Shift> to change starting point xo. L is the Lyapunov exponent calculated for shown finite orbit segment.
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updated 3 Nov 06