The search to understanding chaos is the search to be able to apply a precise mathematical definition to systems which seem on the surface to be undefinable.

Setting aside the complicated mathematics that those who study such things are prone to include in any true explanation of chaos theory, one may find it helpful to begin with what the world looked like before chaos was ever understood.

The Clockwork Universe and Determinism

In the beginning there was Isaac Newton, the laws of gravity and motion, and what was appropriately called a "clockwork universe." In this universe, and under these basic, easy to understand (even for non-scientists) axioms - everything that happened was within the realm of human understanding.

In fact, with these well-tested and rarely questioned laws, it was realized that the universe appeared to be nearly deterministic. Put another way, if one was sufficiently talented and intelligent, they could potentially understand every aspect of the universe and predict exactly what will happen next in any situation.

One could look at the individual particles making up a human brain and understand exactly what the person was thinking, why they were thinking it, and what they were thinking next, based on simple observations regarding the physical occurrences within the gray matter itself. And this principle, in essence, held true for the entire universe.

It was as if once the universe was put into place (whether by natural or supernatural means), it was set into motion, but the future was already entirely determined by the initial conditions of the system. Nothing can possibly happen different, because it is all based on very specific laws.

20th Century Revolutions

Everything changed in the twentieth century.

With the advent of quantum mechanics, it began to grow clear to many scientists that things might not be as deterministic as they seemed. The tiniest particles did not seem to obey any known laws of physics at all, but required the creation of entirely new laws, based not in cause/effect determinacy, but in statistical analysis... "guessing."

The beginning of the twentieth century also brought with it some scientists who began looking into a concept that would later be known as "chaos theory," which, while not entirely closing the book on determinacy, certainly made it clear that using standard laws to make predictions about the natural world was nowhere near as simple as it had seemed.

Defining Chaos

The principle is simple: In any given long-termed system, there will inherently exist slight conditional changes which which amplify exponentially as the process continues, creating a future system which could not possibly have been determined by the initial conditions.

Physicists often use billiard balls as analogies when discussing cause and effect, so it seems appropriate to use them here as well, becuase they can be very useful in defining chaos.

Imagine setting up a simple shot on a pool table - cue ball into the eight-ball. Corner pocket.

With decent conditions on the table itself, and with sufficient talent, this shot is very predictable. But what happens when the shot is complicated: Cue ball into the eight-ball, into the two-ball, into the six-ball. Side pocket.

Suddenly, this becomes a much more difficult shot... but again, with sufficient skill and knowledge of physics, it is not impossible. The problem is, however, that as each progressive action in this process occurs, the "margin of error" becomes slimmer and slimmer.

Even the tiniest crumb of food on the table, or a dent in the felt can change everything - and as more and more complicated shots are considered, even the tiniest error can mean the difference between making the shot and missing by a wide margin.

If the possible routes taken by the cue ball were graphed (in what is called a "bifurcation diagram"), one would see that in the beginning, the divergence was very small, but led to further and further divergences until eventually, the entire system devolved into utter... chaos.

The Butterfly Effect

This, in essence, is what has been known as "The Butterfly Effect" - The notion that a butterfly flapping its wings on one side of the planet will cause a tiny fluctuation in the air around it, which will cause increasingly greater and greater meteorological complications until finally, it all culminates in a hurricane on the other side of the planet at some point in the distant future.

It is ideas like this, which are not entirely without the realm of possibility, which have spurred mathematicians, physicists, climateologists, astronomers and any number of other forms of scientists toward developing a thorough system of comprehending the "incomprehensibility" of chaos - of using mathematics and computer modeling to find patterns within these systems.

Perhaps science will never again propose anything as presumptious as a "clockwork universe," but it continues to strive toward that goal, unreachable as it may be.

References:

Gleick, James. “Chaos: Making a New Science.” Penguin, 1988

Elert, Glen. "The Chaos Hyptertextbook."