It appears e.g. in dynamics of biological populations. 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.

To plot the first iteration we draw vertical red line from the
starting point x toward the blue curve _{o} = 0y = f(x) =
x, where ^{2} + cy.
To get the second iteration we draw red horizontal line to the green
diagonal _{o} = f(x_{o})y = x, where x. Then draw again vertical line to the blue curve to get
_{1} = y_{o} =
f(x_{o})y and so on.
_{1} = f(x_{1})Points f for some _{c}: x_{o} → x_{1} →
x_{2} → ...c and x values
make _{o}orbit of the point x (it is plotted in the right
part of this applet).
_{o} |

The first derivative of a map at a fixed point

is called

So a fixed point is

To the left on complex plane are shown: fractal basin of attractor
z (the blue region), repeller _{1}z and
two points of unstable complex cycle _{2}z.
_{3 }, z_{4 } |

For C = -1 the map has attractin period-2 cycle (the left picture
above). The second iteration of the map f
has two attracting fixed points ^{ o2}z and _{3}z.
The basin of the cycle is shown to the left. _{4 }f
attract the red regions to ^{ o2}z and the blue ones to
_{3 }z. _{4 }z becomes a repeller.
_{1} |

For

Then the Lyapunov exponent is determined as

L

For a chaotic orbit

You see below chaotic quadratic map for

For attracting cycle below *L* is negative

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