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A wellknown example from the study of discrete
dynamical systems is the logistic map. The map
is defined by , where .
The points x=0 and are both fixed points
of the map. Unlike the linear map,
calculating the Lyapunov exponent is not trivial
since the derivative
is not constant for the map. We can, however, do a few
computational experiments to better understand this map.
The following links call up the Maple Form Interface. All
the examples considered involve the case b=4, so
. Please read the
explanations in this document before calling up
the Maple Form Interface so that you can be clear
on what each section of code is actually doing.
(Of course, if you are familiar with Maple,
you can decipher the code easily.)

This bit of code calculates the first N points
in the orbits of x0 and y0.
Initially, N=10 and which
is a fixed point of this map. What happens to the
orbit of y0 when y0 is close to ?
Try changing x0 and y0 to see how orbits of
other neighbouring points behave. (If you increase N,
bear in mind that the output may not be easy to read.)

This code calculates the finite time analog to
the Lyapunov exponent of an orbit orb(x0) over
N iterations. Initially, and
N=50. Increase N to make a reasonable guess
as to what the Lyapunov exponent is.
Change x0 to 0, 0.75 and other values in [0,1]
to see how the approximation
varies.

This code works out how errors in
the initial condition are propagated. Given
x0 and , Maple returns the number
of iterations required for the separation between
orb(x0) and orb() to be multiplied
by 10. Initially, the values and
are used; try changing them. Is
there any pattern emerging?
Contents
Next: Example III: The
Up: Lyapunov Exponents
Previous: Example I: The