Fig.1 illustrates stretching and folding transformations for the quadratic maps
f (for example the Myrberg-Feigenbaum point _{c}c =
-1.401155 is chosen). The segment I is mapped into itself (here _{c} = [-x_{2 },
x_{2}]x is the right repelling fixed point). Points outside
_{2} = 1/2 + (1/4 -
c)^{1/2}I go to infinity.
We see that after one application of _{c}f , there are no
points in _{c}[-x. The segment _{2 }, c)(c is stretched every iteration. Points leave it and never
return back. Thus eventually all points from ^{2}+c,
x_{2}]I come into
_{c}[c, c attractor, bounded by the ^{2}+c]g and _{1}(c) =
cg curves.
_{2}(c) = c^{2}+c |

It maps the interval

After substitution

For

For

It is chaotic tent map. Therefore quadratic map for

Chaotic tent map (as like as the sawtooth map) has the uniform density

The average expansion along a chaotic orbit for

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

For

Then the Lyapunov exponent is determined as

L

For a chaotic orbit

As since for quadratic maps

You see below chaotic quadratic map for

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