This article is from the Nonlinear Science FAQ, by James D. Meiss firstname.lastname@example.org with numerous contributions by others.
(Thanks to Bruce Stewart for Contributions to this answer)
In order to address this question, we must first agree what we mean by chaos,
In dynamical systems theory, chaos means irregular fluctuations in a
deterministic system (see [2.3] and [3.7]). This means the system behaves
irregularly because of its own internal logic, not because of random forces
acting from outside. Of course, if you define your dynamical system to be the
socio-economic behavior of the entire planet, nothing acts randomly from
outside (except perhaps the occasional meteor), so you have a dynamical
system. But its dimension (number of state variables--see [2.4]) is vast, and
there is no hope of exploiting the determinism. This is high-dimensional
chaos, which might just as well be truly random behavior. In this sense, the
stock market is chaotic, but who cares?
To be useful, economic chaos would have to involve some kind of collective
behavior which can be fully described by a small number of variables. In the
lingo, the system would have to be self-organizing, resulting in low-
dimensional chaos. If this turns out to be true, then you can exploit the low-
dimensional chaos to make short-term predictions. The problem is to identify
the state variables which characterize the collective modes. Furthermore,
having limited the number of state variables, many events now become external
to the system, that is, the system is operating in a changing environment,
which makes the problem of system identification very difficult.
If there were such collective modes of fluctuation, market players would
probably know about them; economic theory says that if many people recognized
these patterns, the actions they would take to exploit them would quickly
nullify the patterns. Market participants would probably not need to know
chaos theory for this to happen. Therefore if these patterns exist, they must
be hard to recognize because they do not emerge clearly from the sea of noise
caused by individual actions; or the patterns last only a very short time
following some upset to the markets; or both.
A number of people and groups have tried to find these patterns. So far the
published results are negative. There are also commercial ventures involving
prominent researchers in the field of chaos; we have no idea how well they are
succeeding, or indeed whether they are looking for low-dimensional chaos. In
fact it seems unlikely that markets remain stationary long enough to identify
a chaotic attractor (see [2.12]). If you know chaos theory and would like to
devote yourself to the rhythms of market trading, you might find a trading
firm which will give you a chance to try your ideas. But don't expect them to
give you a share of any profits you may make for them :-) !
In short, anyone who tells you about the secrets of chaos in the stock market
doesn't know anything useful, and anyone who knows will not tell. It's an
interesting question, but you're unlikely to find the answer.
On the other hand, one might ask a more general question: is market behavior
adequately described by linear models, or are there signs of nonlinearity in
financial market data? Here the prospect is more favorable. Time series
analysis (see [3.14]) has been applied these tests to financial data; the
results often indicate that nonlinear structure is present. See e.g. the book
by Brock, Hsieh, LeBaron, "Nonlinear Dynamics, Chaos, and Instability", MIT
Press, 1991; and an update by B. LeBaron, "Chaos and nonlinear forecastability
in economics and finance," Philosophical Transactions of the Royal Society,
Series A, vol 348, Sept 1994, pp 397-404. This approach does not provide a
formula for making money, but it is stimulating some rethinking of economic
modeling. A book by Richard M. Goodwin, "Chaotic Economic Dynamics," Oxford
UP, 1990, begins to explore the implications for business cycles.