Fig.1 illustrates stretching and folding transformations for the quadratic maps fc (for example the Myrberg-Feigenbaum point c = -1.401155 is choosed). 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. You see below intervals Ic (the left picture) and chaotic attractors (the right one) for different c values.

# Chaotic dynamics for c = -2

Let us consider quadratic map with 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 of x = 2 cos(2p y) we get
cos(2p yn+1) = 2 cos2(2p yn) - 1 = cos(4p yn) .
The sawtooth map yn+1 = 2yn (mod 1) is a solution of this equation. Therefore quadratic map for c = -2 has infinite countable set of unstable periodic and continuum set of chaotic orbits. One of period-3 orbits is (we should use the tent map to get both periodic orbits)
x0 = 2 cos(2p 1/7) = 1.24698,     x1 = 2 cos(2p 2/7) = -0.445042,
x2 = 2 cos(2p 4/7) = -1.80194
.

From x = 2 cos(2p y) one gets
|dx| = 4p |sin(2p y)| dy = 2p (4 - x2)1/2 dy .
For the sawtooth map with the uniform density relative number of points of a chaotic orbit in a small interval dy is r(y) dy = dy. As since all these points are mapped in interval dx, therefore the number is equal to r(x)|dx| for our quadratic map and corresponding invariant density is (there is 2 below because y is mapped in x twice)
r(x) = 2 dy / |dx| = 1/p (4 - x2)-1/2 .
 The density is shown to the left in blue-green-red colors. The average expansion along a chaotic orbit for c = -2 is     ò |2x| r(x) dx = 8/p = 2.54648 .

It seems very likely that one will get similar analitical densities for band merging and interior crisis points (two of them are shown above).

You see below more complicated chaotic orbits and invariant densities

# The basic dichotomy for real maps

For almost every c Î [-2, 1/4], the quadratic map fc: x ® x2 + c is either regular or stochastic.

For quadratic maps it is proven that the set of c values for which attractor is chaotic has positive Lebesque measure and attracting periodic orbits are dence in the set, i.e. between any two chaotic parameter values there is always a periodic interval [1]. We will see these amazing structures on the next page.

[1] Mikhail Lyubich The Quadratic Family as a Qualitatively Solvable Model of Chaos Notices of the AMS, 47, 1042-1052 (2000)

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updated 28 December 2002