Dynamical systems, orbits and attractors

 

A dynamical system is a mapping of a metric space into itself. To lend some control to the ``generations,'' we require that our mapping be continuous. Continuity may be defined entirely in terms of limits. Suppose is a sequence of points in X that has a limit . In any such situation, we insist that

If this is always the case, we declare F to be continuous.

L-systems have such a continuous mapping associated to them. For example, in the case of the Fibonacci L-system, we define

and for any trajectory we define . We leave it as an exercise to prove that F is actually continuous.

The continuous mapping produces the generations by iteration. Starting with the initial generation or state , we define , , etc. Briefly, we write , and we call the function obtained by composing F with itself n times the n-th iterate of F. The sequence is called the orbit of under F.

For the Fibonacci L-system, we have seen that

the Fibonacci sequence of a's and b's. It can be shown that this limit is true no matter what the initial generation is chosen to be in code-space. The limit sequence satisfies ; we say is a fixed point of F. In fact, is the only fixed point of F. is an example of an attractor of a dynamical system. Attractors may be much more complicated than a single point; all the fractals we shall consider are attractors of some relatively simple dynamical systems, and this is a large part of their interest.

Attractors arise as the limiting behavior of orbits. Sequences can have tremendously variable behavior.

• The sequence might have a limit.

  

• The sequence may be unbounded:

  

  (The distance between and grows larger and larger without bound.)

• Part of the sequence may tend to one limit point, and others to other limit points. For example, we might have

  

  (The even terms tend to y; the odd terms to z.)

This leads to the following notions.

subsequence:

A subsequence of a sequence is a selection where . For instance, we could select , , , etc., and obtain the subsequence of even-indexed terms. There are infinitely many choices of possible subsequences.

point of accumulation:

A point of accumulation, also called a limit point, of a sequence is a point that occurs as the limit of some subsequence . That is,

Take special note of the use of ``a'' instead of ``the.'' A sequence can have lots of points of accumulation.

limit set:

The limit set of a sequence is the set of points of accumulation of the sequence.

attractor:

An attractor of a dynamical system is the limit set of some orbit . Different orbits may have different attractors. It's a particularly nice situation (and not uncommon) when the attractor is the same for all orbits. That's the case for the Fibonacci L-system.

When does a sequence have a point of accumulation? That depends on possible ``holes'' in our metric space. One common assumption is that the metric space has no ``holes,'' other than possibly. That leads to two more popular pieces of terminology

complete:

A metric space is complete if every bounded sequence has at least one point of accumulation.

compact:

A metric space is compact if every sequence (bounded or otherwise) has at least one point of accumulation.

The same terms may be applied to subsets of a metric space that meet the criteria. Euclidean space is known to be complete (but not compact); code-space is actually compact.