INTRO TO CHAOS THEORY

Tavishi

Chaos theory has no one perfect definition, (fitting I suppose), but if I was to put it simply, chaos theory is the mathematical study of complexity that somehow extends to every possible field. Chaos theory seeks the patterns within something seemingly pattern-less, trying to find some order within disorder. Chaos theory studies complex systems with more than one component, like a haul-out of rowdy California sea lions (to which the colloquial definition of chaos can be assigned...).

We are going to say these sea lions have predictable behavior; if the sun continues to be warm, they will continue to sleep here. (There's a lot more complexity to this system but I wanted to use seals as an example.) We are going to assume this system of sea lions on the pier is deterministic (key word assume). This means that for every set input, there is a given, fixed output. If the sun continues to be warm, and the pier drifts a little, the sea lions will continue to rest on the piers and sleep, no matter what. If someone riles up the sea lions, they will bark, no matter what. If the sun sets, the sea lions will all dive into the sea, no matter what. These inputs are entirely responsible for the outputs (in this case, the sea lion behavior.) Deterministic systems are one of the key assumptions of chaos theory.

Chaos, however, emerges in a deterministic system when studying how even the slightest change in input results in an extremely different outcome in the end. For example, if one of the sea lions accidentally falls into the water, causing a splash, the entire herd may emerge into unpredictable behavior. The rest of the group may flee, fearing a predator, or just generally engage in entirely different behavior than would happen if that one sea lion had not fallen into the ocean. This is known as deterministic chaos.

If you've ever watched Star Trek, this sort of guiding principle is the reason for the Prime Directive. Or, a more well-known well-discussed theory: the Butterfly effect, which was actually derived from Edward Lorenz's work. Chaos theory studies the patterns among this seemingly unpredictable chaos. It searches for linkages between the "random" behavior in a deterministic system and the initial conditions. What's beautiful to me about chaos theory is how it seems to connect to so many fields, while still being so deeply mathematical. While we've roughly defined chaos, if we take a more quantitative approach to defining chaos, we can say that deterministic chaos is sensitive to initial conditions, topologically transitive, and has dense periodic orbits. This is a lot of word vomit which I will absolutely explain further. First, sensitivity to initial conditions. As discussed earlier, when that tiny change in one sea lion's behavior occurred, a completely different outcome resulted. Or similarly, the Butterfly effect, or the effects of time travel.

If we put this into math terms, we look at the sensitivity to initial conditions. We want to identify two different trajectories; one that has experienced the change in condition, and one that is the "original" condition. We can find the rate of change of separation between these trajectories by multiplying the initial separation of the trajectories with e to the λt power.

λ is the Lyapunov exponent, or the measure of the sensitivity to initial conditions. This is a pretty difficult concept to explain well, and I am currently struggling with fully grasping it myself, but I am more than open to making an entire Kradigan just for λ.

Next, topologically transitivity and dense periodic orbits. A dynamical system, in which we study chaos, is one that measures a function dependent on time. In the example above, we can track the pinnipeds' motion over time.

Topological transitivity is a difficult concept to explain without too much pure math. If we take a dynamical system and take two open sets S and L, the sets will coincide at some point, meaning it is impossible to decompose a dynamical and topologically transitive system into two different sets. There will always be some overlap. Periodic orbits result from repeatedly applying the function f (essentially raising f to the kth power), creating an orbit and set of points.

Below is a mathematical representation of topological transitivity:

f^(k)(S) ∩ L ≠ 0

Dense periodic orbits implies that every point in that space within our dynamical system are approached arbitrarily close by some periodic orbit.

Now that we have loosely defined chaos, let's talk more about chaos theory as a whole. As per the sea lion example prior, when that one sea lion falls into the water, all the sea lions freak out and panic and jump off the rafts. Now, if someone runs onto the rafts and startles the sea lions, the sea lions will also freak out, panic, and jump off the rafts. If a school of tasty, tasty fishies swim just below the rafts, many of the sea lions will jump into the water, startling the others into jumping in, too.

While it may seem like pure randomness, these initial conditions seem to lead to one very similar outcome/path. An attractor is an outcome toward which a set of states tends to evolve. In math terms, an attractor is a subset C (c is for ceal) for which a basin of attraction exists. Within this basin of attraction, which is just a neighborhood of C, all points x (x is for xeal) exist within C at the limit t (this is a function of time, remember) goes to infinity.

Attractors, however, differ a little within chaotic systems.

A chaotic attractor, better known as a strange attractor, forms a fractal pattern. A fractal is a geometric pattern with an insane level of complexity (you can make shape super super tiny!) This is measured with the Hausdorff dimension- an integer value for the Hausdorff dimension would be something simple like a cube (Hausdorff dimension of 3). A fractal, on the other hand, would have a Hausdorff dimension of something like 6.96. One of the best known strange attractors is a Lorenz attractor:

The good ol' Lorenz attractor is a series of differential equations called the Lorenz equations. If you are to look at the path of a particle on a Lorenz attractor for a series of particular solutions, it's so interesting to watch how easily a small change in the initial conditions may lead to an entirely different solution. Other strange attractors include the Henon, Rossler, and double-scroll attractors.