# Quadratic map

More complicated analytic quadratic map is

*x*_{n+1} = f_{c}( x_{n} ) =
x_{n}^{2} + c.

On complex plane it
generates the famous Mandelbrot and Julia fractal sets. In spite of apparent
simplicity it has very rich dynamics. For this map regions of regular and
chaotic dynamics are entangled in an intricate manner and scenarios of
transition to chaos are common for many other dynamical systems.
## Iteration diagram

Dynamics of 1D real maps is useful to trace on iteration diagram
shown below. The blue curve is
*f*^{ oN}(x) = f(f(...f(x))) the *N*-th iteration
of *f(x)*. Diagonal *y = x* is the green
line. *-2 ≤ x,y ≤ 2*. As since *f(0) = C* then for *N = 1*
the *C* value coincides with *y(0)*.
Dependence *x*_{n} on *n* is plotted in the right window.
*Controls*: Drag the blue curve to change *C* and
starting point *x*_{o} values.
Press <Enter> to set new parameters from the text fields.

To plot the first iteration we draw vertical red line from the
starting point *x*_{o} toward the blue curve *y = f(x) =
x*^{2} + c, where *y*_{o} = f(x_{o }).
To get the second iteration we draw red horizontal line to the green
diagonal *y = x*, where *x*_{1} = y_{o }.
Then draw again vertical line to the blue curve to get
*y*_{1} = f(x_{1 }) and so on.

Points *f*_{c}: x_{o} → x_{1} →
x_{2} → ... for some *c* and *x*_{o} values
make *orbit* of the point *x*_{o} (it is plotted
to the right).
## Critical points

For an analytic map points where *f '(x*_{c }) = 0 are called
*critical points*. Every stable cycle attracts at least one critical
point. Quadratic map has the only critical point *x*_{c} = 0.
Therefore it can have only one attracting cycle and *x*_{c} is
used as the starting point to find the cycle.
## Fixed points

For *C = -1/2* iterations go quickly to attracting
*fixed point x*_{•} = f(x_{• }) of the
map. Fixed points correspond to intersections of *y = x* and
*y = f(x)* (green and blue) curves. There are always two fixed points
(may be complex) for a quadratic map because of
two roots of quadratic equation

*f(x*_{•}) - x_{•} =
x_{•}^{2} + c - x_{•} = 0,
x_{1,2} = 1/2 ∓ (1/4 - c)^{½}.

The first derivative of a map at a fixed point

*m = f '(x*_{•}) = 2x_{•}

is called *multiplier* (or the *eigenvalue*) of the point.
For small enough *δx*

*f(x*_{•} + δx) =
f(x_{•}) + mδx +
O(δx^{2}) ≈ x_{•} + mδx.

So a fixed point is *stable* (*attracting*),
*superstable*, *repelling*, *indifferent* (*neutral*)
according as its multiplier satisfies *|m| < 1*,
*|m| = 0*, *|m| > 1* or *|m| = 1*.
The second fixed point (the right intersection) is always repelling. For
*|x| > x*_{2} iterations go to infinity. For
*|x| < x*_{2} they go to the attracting fixed point
*x*_{1}. This interval is the *basin of attraction* of
the point.

## Attracting cycles

For *C = -1* the map has attractin period-2 cycle (the left picture
above). The second iteration of the map *f*^{ o2}
has two attracting fixed points *z*_{3} and *z*_{4 }.
## Lyapunov exponent

For a continuous map *x*_{n+1} = f(x_{n })
a small deviation *δx*_{o} of coordinate
*x*_{o} leads to a small change in *x*_{1}

*δx*_{1} =
f '(x_{o}) δx_{o}.

For *n* iterations

*δx*_{n} =
δx_{o}∏_{ i=0,n-1} f '(x_{i }).

Then the Lyapunov exponent is determined as

* Λ = lim*_{n → ∞} L_{n} ,

L_{n} = 1/n
log|δx_{n }/δx_{o}| =
1/n ∑_{i=0,n-1} ln |f '(x_{i })|.

For a chaotic orbit *|δx*_{n}| grows with
increasing of *n* so *Λ > 0*.

You see below chaotic quadratic map for *c = -2* with positive
Lyapunov exponent *L* calculated for shown finite orbit segment.
For attracting cycle below *L* is negative

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*updated* 3 July 2007