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 = [-x₂, x₂] is mapped into itself (here x₂ 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 [-x₂, c). The segment (c²+c, x₂] is stretched every iteration. Points leave it and never return back. Thus eventually all points from I_c come into [c, c²+c] attractor, bounded by the g₁(c) = c and g₂(c) = c²+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 ◅ 2² ◅ 2³ ◅ ... ◅ 2²7 ◅ 2²5 ◅ 2²3 ◅ ... ◅ 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_n² - 2 .
It maps the interval [-2,2] onto itself. After substitution x = 2 cos(πy) we get
cos(πy_n+1) = 2 cos²(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 - x²)^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 - x²)^-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