Pardon my quoting, I'm very new to this.
I liked the idea when it was first presented (to me and millions of others) in the Jurassic Park movie - but never looked into it except briefly.
When I'm attempting to learn a new subject I find it useful to define elementary terms but in this case there seems that nothing is 'elementary', hence the quotes.
Quote:
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CHAOS THEORY AND DATA ANALYSIS
Introduction to Chaos
Essentially, chaos is a
nonlinear behavior that exists between the realms of
periodic and
random. At first glance, some chaotic systems may appear to regular and periodic, whereas
others will appear strictly random; in both cases closer examination topples these assumptions.
Strictly speaking, chaotic systems are deterministic and, the exact system state can be written:
X(t) =(x(t),x(t −τ ),x(t − 2τ ),...,x(t − (k −1)τ)
- where t is a scalar index for the data series and τ is the interval of observations.
Let F: ℜk →ℜk be the nonlinear function governing the system; then, the future state of the system at any time t+τ can be ascertained.
However, as no real-world system is likely to be completely deterministic, a (relatively small) probabilistic component, p(t), with mean zero is added to account for random effects (Lu and Smith, 1991): x(t+τ) = F(X(t)) + p(t).
- The state of the system, X(t), is critical to knowing the progression of a system, and even
a small change in it will radically alter the manner in which the system evolves. Thus, after a
short interval, the system effectively becomes unpredictable. This effect is known as sensitivity to initial conditions and is a hallmark of chaotic systems.
I've added emphasis by way of bolding and italics to what I consider the essential parts.
While attempting to further define this I've found it useful to look at what Chaotic Systems are NOT.
Wiki comes in handy here and provides many examples, one of which is called a "Random Walk".
It's easy enough to skim through and get the idea if you're not familiar with the term.
Since I like probability it was familiar:
Wiki Random Walk.
We need to examine 'change states' in order to determine if each process is stochastic (random) the counterpart to deterministic processes.
I was already familiar with Brownian Motion so that helped me a bit.
The Intro to Chaos (quoted above) goes on to show how "dissipative¹, nonrandom system eventually will settle onto a path called an attractor."
Understanding of 'attractors' is also essential to basic understanding of the theory.
My next 'step' is to try to find a good visualization of an analysis of a chaotic system and post it here. It actually helps me quite a bit to try to 'teach' when I'm learning because it forces me to fix my thoughts in such a way that they can be communicated. For those of you who don't appreciate my method I'd suggest that you follow your own path and add to the discussion here as others have done
I'm stretching quite a bit here so (it was much easier to learn new things when I was younger) and again, pardon if it doesn't make complete sense yet.
~Granps
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EndNotes, Annotations & Credits
Quote 1:
CHAOS THEORY AND DATA ANALYSISFootnote¹ A
dissipative system is essential one that requires a constant supply of energy to keep running. Thus, such a
system will, over time, attempt to move towards an equilibrium in terms of energy (i.e. an attractor).
Best 'Intro' that I've found so far: [Continue] 
