What is Chaos Theory?
Chaos Theory is the means to study the behavior of dynamic systems. Dynamic systems are different from static systems due to the sheer fact that static systems are non–changing, whereas dynamic systems are changing. A pure cybernetic system can be classified as a static system, e.g. a clock or oscillating fan. These pure cybernetic systems are designed with all their component interactions predicted and calibrated during its architectural and manufacturing phase. Dynamic systems are systems that change, either form endogenous factors or exogenous factors. Some dynamic systems can be very complex, e.g. the human body—while other dynamic systems are quite simple, such as the unfolding dynamics of a salt crystal with water.
Dynamic systems are nonlinear systems. These nonlinear systems can be unpredictable and can possess amplifying feedback loops. In terms of chaos theory, there are deterministic chaos and non–deterministic chaos. Deterministic chaos is simple chaos, whereby no stochastic functions are in the system. Examples of deterministic chaotic systems are Lorenz Systems and double fulcrum pendulums.
In deterministic chaotic systems, though there are no stochastic perturbations in the system, there are unpredictable behaviors exhibited in these systems. In a typical Lorenz system there are two chaotic attractors. One can predict an overall trend within the Lorenz system but cannot provide exact position. In the double fulcrum pendulum example, all the mechanics of this system can be expressed in a mathematical formula—but the design of this system will still exhibit nonlinear behavior. When the double fulcrum pendulum moves back-and-forth with high velocity the two fulcrums start to exhibit their degrees of the free, whereby the lower level can move erratically in a circular pattern.
In non–deterministic chaotic systems there are complex chaos. Complex chaos is where there are stochastic functions present in the system. Examples of non–determinist chaotic systems are social systems and financial markets. Another interesting complex chaotic system is the evolution of a particular war. In battle, due to the high degrees of freedom on the field, the conflict can evolve in unexpected ways. The evolution of war is quite similar to the evolution of weather, whereby there are feedback loops that change the present dynamics of the system leading to unpredictable behavior.
In terms of financial markets, human misbehavior, random news events, and feedback loops contribute to the changing dynamics. Behavioral finance articulates these forces on the market to help explain bubbles and crashes. In Soros (2003), reflexivity is a process in which self–reinforcing processes produce disequilibrium in financial markets.
When modeling financial markets there is a degree of error, which is quite different when modeling the motion of objects with Newton’s Laws. Why is it that a financial model is less accurate than models in physics? The basic answer is that the laws of physics do not change—they are stationary in a sense. But financial dynamics are very complex and have changing attributes, similar to history. History does not repeat, but it rhymes. Within this rhythm of history are the stochastic perturbations that are forceful enough to make the system different. We call this a path dependency, whereby the acts of the past and current states affect future states. In quantitative finance we call this path dependency a Markov chain when only the present state is affecting future states.