Q3a: What is chaos?
A3a: Chaos is apparently unpredictable behavior arising in a deterministic system because of great sensitivity to initial conditions. Chaos arises in a dynamical system if two arbitrarily close starting points diverge exponentially, so that their future behavior is eventually unpredictable.
Weather is considered chaotic since arbitrarily small variations in initial conditions can result in radically different weather later. This may limit the possibilities of long-term weather forecasting. (The canonical example is the possibility of a butterfly's sneeze affecting the weather enough to cause a hurricane weeks later.)
Devaney defines a function as chaotic if it has sensitive dependence on initial conditions, it is topologically transitive, and periodic points are dense. In other words, it is unpredictable, indecomposable, and yet contains regularity.
Allgood and Yorke define chaos as a trajectory that is exponentially unstable and neither periodic or asymptotically periodic. That is, it oscillates irregularly without settling down.
sci.fractals may not be the best place for chaos/non-linear dynamics questions, sci.nonlinear newsgroup should be much better.
Q3b: Are fractals and chaos synonymous?
A3b: No. Many people do confuse the two
domains because books or papers about chaos speak of the two concepts or are
illustrated with fractals.
Fractals and deterministic chaos are mathematical tools to
modelise different kinds of natural phenomena or objects. The keywords in
chaos are impredictability, sensitivity to initial conditions in spite of
the deterministic set of equations describing the phenomenon.
On the other hand, the keywords to fractals are self-similarity, invariance of scale. Many fractals are in no way chaotic (Sirpinski triangle, Koch curve...).
However, starting from very differents point of view, the two domains have many things in common : many chaotic phenomena exhibit fractals structures (in their strange attractors for example... fractal structure is also obvious in chaotics phenomena due to successive bifurcations ; see for example the logistic equation Q9 )
The following resources may be helpful to understand chaos:
Q3c: Are there references to fractals used as financial models?
A3c: Most references are related to chaos being used as a model for financial forecasting.
One reference that is about fractal models is, Fractal Market Analysis - Applying Chaos Theory to Investment & Economics by Edgar Peters.
Some recommended Chaos-related texts in financial forecasting.