December 14th, 2005, 07:10 PM
I will try to relate those terms. The following is nothing you can read
Are complex systems nothing more than a system of chaotic equations?
Similiar to the system of equations used in linear algebra and matrices
in a book or article. It is so to say my interpretation of things I have
read. I also checked the Wikipedia-articles, which are okay. I first try
to be mathematical, then I give some "intuitive" interpretation or examples.
complex systems are given by a number of entities (variables x_1(t),x_2(t), ...),
which are connected non-linearly, e.g. d x_1(t) / d_t = -x_1(t)^2 * x_2(t)... plus
similar equations for x_2(t) etc, where "t" can be interpreted as the time.
Note that if you set x_2(t)=x_3(t)=...=1 you will recover something like a marginal
version of the logistic map (just discretise time).
The study of those complex systems is called system theory.
Mathematically, one is interested in two different things: fixed points and dynamics.
- A fixed point is a set of specific values of x_1(t), x_2(t), such that they do not
change while further evolving the system in time.
- Then, the dynamics could be such that certain values of x_1(t), x_2(t) etc. are
repeated periodically (or in another similarity pattern).
- Further, the dynamics could be that no such similiarity pattern can be found. Then,
the complex system is called "chaotic".
Since the dynamics of complex systems is difficult to control, one may
perform simulations. Typically, so-called multi-agent systems/cellular automata
are used. Why?
A complex system can be understood as a set of individuals, who interact
with each other. Thus, the reference to the "Game of Life" above. They may
adapt their strategy (called complex adaptive systems) or not.
For example, Psychologists have invented a "multi-agent system" called the
"Cybernetic model": since one cannot "calculate" why certain animals form
certain "societies", one invented a model, which controls the movements of
an animal based on some interactions with the environment. Thus, the way
how animals herd, allows to extract information about the "character" of
the animals. The same, in some limits, is also possible for human beings.
For example, Blackjack players (me ) have tried to model interaction to
extract the optimal game strategy. In the end, several millions of games have
been played using computer simulations.
For example, the stock market: shared/derivatives etc. show a certain time history,
which can be analysed. Typical properties of complex (adaptive) systems, like memory
and "fat tails" in the distribution can be seen. All this can be interpreted in
the context of a "multi-agent system".
It even can be a way to analyse "network traffic", and hence, security: you have
entities, which interact with each other. Having a "baseline", you may detect deviations.
In some context of interpretation, we here even may talk of a reference monitor.
If the only tool you have is a hammer, you tend to see every problem as a nail.
(Abraham Maslow, Psychologist, 1908-70)