Our first example will be the Fibonacci L-system with the following components
The fascinating behavior happens when we set an L-system in motion,
evolving from moment to moment. The evolution of an
L-system is defined as a sequence 
, 
where each generation 
is a word in 
that evolves
from the previous generation 
by applying all the production
rules to each symbol in . The first generation 
is the
axiom 
. The first few generations of the Fibonacci system are
as follows:
|
| a |
 | b |
 | ba |
 | bab |
 | babba |
 | babbabab |
 | babbababbabba |
 | babbababbabbababbabab |
We may think of the individual a's and b's as life-forms, and the productions as stages in their lives. After one generation, the ``immature'' life-form a matures into an adult b. After that the adult b is able to produce one baby a each generation. Thus, this symbolic dynamical system models a very simple kind of population dynamics.
The origin of the name ``Fibonacci'' comes from the connection with a
special sequence of numbers 
known as the
Fibonacci numbers. These are defined by a second-order recurrence
relation
You may check for the few generations listed above that the number of
organisms in the n-th generation is 
. Why this
should be true is a mystery until we think about it further. The
reason is that this dynamical system, defined by a local
process affecting each individual organism (the production rules),
exhibits a global process at work at the same time, defined in
terms of whole populations. After staring at the generations, we see
the pattern:
by which we mean that generation n+2 consists of generation n+1 followed by generation n, for example:
As an exercise, we ask the reader to supply a proof of this global process.