LogisticMap
The LogisticMap model shown using Vergil, the Ptolemy II schematic editor:
This model shows a polynomial mapping described by
the equation: x(n+1) = rx(n)(1-x(n)) which was
originally introduced as a demographic model where
x(n) is a number between 0 and 1, and represents
the population at year n and r is a positive number,
and represents a combined rate for reproduction and
starvation.
It is often cited as an archetypal example of how
complex, chaotic behavior can arise from very simple
non-linear dynamical equations:
- With 0, the population will eventually die.
- With 1, the population will quickly stabilize to
(r-1)/r.
- With 2<r<3, the population will eventually stabilize
to (r-1)/r, after oscillating around that value for
some time.
- With 3<r<3.45, the population will oscillate between
2 values forever.
- With 3.45<r<3.54, the population will oscillate
between 4 values.
- With r slightly bigger than 3.54, the population
will oscillate between 8 values, then 16, 32, etc.
The lengths of the parameter intervals which yield
the same number of oscillations decrease rapidly...
- At r = 3.57 is the onset of chaos and we can no
longer see any oscillations.
For more details, see
http://en.wikipedia.org/wiki/Logistic_map