# Chaotic quadratic maps

Fig.1 illustrates stretching and folding transformations for the quadratic maps
*f*_{c} (for example the Myrberg-Feigenbaum point *c =
-1.401155* is chosen). The segment *I*_{c} = [-x_{2 },
x_{2}] is mapped into itself (here *x*_{2}
is the right repelling fixed point). Points outside
*I*_{c} go to infinity.
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.

## "Period three implies chaos"

In 1975 T.Y.Li and J.A.Yorke published the famous "Period three implies
chaos" paper. It turns out that nonlinear 1D map with period-3 orbit has
continuum of chaotic orbits. In short, let *a,b,c* make period-3 cycle

*f(a) = b, f(b) = c, f(c) = a*.

It follows from the Fig.2 that inverse function *f(x)*^{-1}
is multivalued in *(b,c)*. When iterated *f(x)*^{-1}
value gets in this interval we can chouse any branch at random and make
chaotic orbits.

## Sharkovskii's theorem

Moreover if a map has period-3 orbit then it has orbits with every period.
It is particular case of Sharkovskii's theorem that a map with period-*n*
orbit has orbits with all periods *n'* preceding *n* in the list

*1 ◅ 2 ◅ 2*^{2} ◅ 2^{3} ◅ ...
◅ 2^{2}7 ◅ 2^{2}5 ◅ 2^{2}3 ◅ ...
◅ 2·7 ◅ 2·5 ◅ 2·3 ◅ ...
◅ 7 ◅ 5 ◅ 3

where *... 7 5 3* are odd numbers.
Stable and unstable period-*p* cycles for quadratic map appear after
tangent bifurcation of the *f*^{op}(x) map. With decreasing
*c* stable cycle loses its stability and two period-*2p*
cycles appear. Unstable cycles never die. Therefore after period doubling
cascade completion quadratic maps have infinite set of unstable periodic
orbits.

## Chaotic dynamics for *c = -2*

Let us consider quadratic maps for *c = -2*

*x*_{n+1} = x_{n}^{2} - 2 .

It maps the interval *[-2,2]* onto itself.
After substitution *x = 2 cos(πy)* we get

*cos(πy*_{n+1}) =
2 cos^{2}(πy_{n }) - 1 = cos(2πy_{n }).

For *y*_{n } ≤ 1/2 it follows

*y*_{n+1} = 2y_{n} .

For *y*_{n } > 1/2 by means of formula
*cos(2πy*_{n }) = cos(2π - 2πy_{n }) we
reduce cosine argument to the interval *[0,π]* and get

*y*_{n+1} = 2 - 2y_{n} .

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 - x*^{2})^{½} 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 - x^{2})^{ -½}.

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.
## Lyapunov exponent

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*.
Change *x*_{o} 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.
*x*_{o} = 0).

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*updated* 3 Nov 2006