We see that after one application of f_{c} , there are no
points in [-x_{2}, c). The segment (c^{2}+c,
x_{2}] is stretched every iteration. Points leave it and never
return back. Thus eventually all points from I_{c} come into
[c, c^{2}+c] attractor, bounded by the g_{1}(c) =
c and g_{2}(c) = c^{2}+c curves.
You see below intervals I_{c} (the left picture) and chaotic attractors (the right one) for different c values. |
From x = 2 cos(2p y) one gets
|dx| = 4p
|sin(2p y)| dy = 2p
(4 - x^{2})^{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 - x^{2})^{-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 . |
You see below more complicated chaotic orbits and invariant densities
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)