## What Is Chaos Theory?

Chaos theory is a mathematical concept that explains that it is possible to get random results from normal equations. The main precept behind this theory is the underlying notion of small occurrences significantly affecting the outcomes of seemingly unrelated events. Chaos theory is also referred to as "non-linear dynamics."

### Key Takeaways

- Chaos theory is a mathematical concept that explains that it is possible to get random results from normal equations.
- Simply put, chaos theory is an attempt to see and understand the underlying order of complex systems that may appear to be without order at first glance.
- The first real experiment in chaos theory was done in 1960 by a meteorologist, Edward Lorenz.
- One of the key concepts of chaos theory is the butterfly effect, which states that a minuscule variation in starting conditions for a model can result in wide variations in the end conditions.
- In finance, the fractal market hypothesis uses the principles of chaos theory to predict the behavior of uncertain markets.

## Understanding Chaos Theory

Chaos theory studies the patterns and regularities that arise from disordered systems. It has been applied to many different things, from predicting weather patterns to the stock market. Simply put, chaos theory is an attempt to see and understand the underlying order of complex systems that may appear to be without order at first glance.

While the origins of chaos theory date back to the 19th century, the development of advanced computational techniques made it easier to study the behavior of complex systems, such as weather and fluid dynamics. While the basic equations for these systems may be relatively simple, a tiny variation in the starting conditions can result in unexpected outcomes.

## History of Chaos Theory

The first real experiment in chaos theory was done in 1960 by a meteorologist, Edward Lorenz. He was working with a system of equations to predict what the weather would likely be.

In 1961, he wanted to recreate a past weather sequence, but he began the sequence midway and printed out only the first three decimal places instead of the full six. This radically changed the sequence, which could reasonably be assumed to closely mirror the original sequence with only the slight change of three decimal places.

However, Lorenz proved that seemingly insignificant factors can have a huge effect on the overall outcome. Chaos theory explores the effects of small occurrences dramatically affecting the outcomes of seemingly unrelated events.

The principles of chaos theory and the butterfly effect were popularized by the fictional mathematician Ian Malcolm in the 1993 film *Jurassic Park.*

## Chaos Theory and the Butterfly Effect

Chaos theory is closely linked to the *butterfly effect,* the idea that a very slight variation in initial conditions can result in substantial changes in output values. This runs counter to the intuitive idea that similar initial conditions should result in roughly similar outputs.

This concept frequently occurs in mathematical modeling, where a tiny rounding error in the initial conditions can result in wildly different outcomes. In popular culture, it is commonly illustrated with the idea that a butterfly flapping its wings in one part of the world could make all the difference between a blizzard and a hurricane in another hemisphere.

## Chaos Theory in the Stock Market

Chaos theory is a controversial and complicated theory that has been used to explain some features of systems that have traditionally been difficult to accurately model. The financial markets fall into this category with the additional benefit of a rich set of historical data. One interesting financial phenomenon that chaos theory can help illustrate, if not explain, is how seemingly healthy financial markets can suffer sudden shocks and crashes.

Proponents of chaos theory believe that price is the very last thing to change for a stock, bond, or other security. This suggests that periods of low price volatility do not necessarily reflect the true health of the market. Looking at the price as a lagging indicator puts investors in the dark as far as being able to spot crashes before they happen.

The fractal market hypothesis suggests that in times of market uncertainty, price movements may move in a fractal pattern rather than a random walk. in other words, the movements that occur on a small time scale may be repeated on a larger scale.

This does, of course, fit the experience of most investors who have experienced black swan events and financial meltdowns. There are some who seem to be able to position themselves for market downturns in advance, but they are often digging much deeper than price data to understand structural weaknesses that most of the market has overlooked.

The big caveat with chaos theory is that it is too often used as a way to discount investing. While the markets are almost impossible to predict over a short-term period, they are more consistent over the long run. Just because you can't time the next crash doesn't mean you shouldn't be investing in stocks with strong fundamentals that tend to perform over the long term.

## Example of Chaos Theory

In finance, the Fractal Market Hypothesis uses elements of chaos theory to predict swings in the stock market. This theory is an extension of the efficient market hypothesis, which suggests that prices move in a random walk.

The fractal market hypothesis states that during times of high uncertainty, price movements can show similar behavior when viewed over different time horizons. This may be used in technical analysis, where repeating or recursive patterns can be used to project future price moves.

## Who Invented Chaos Theory?

The discovery of chaos theory is usually attributed to Edward Lorenz, a meteorologist at the Massachusetts Institute of Technology in 1961. Lorenz was using mathematical models to predict future weather patterns, but he discovered that the predictions could vary widely depending on how many accurately he set the starting conditions. In other words, an extra decimal point of accuracy could result in extremely large variations on the final outcome of the model.

## What Is the Butterfly Effect?

The butterfly effect is a principle of chaos theory that states that certain nonlinear deterministic systems are highly sensitive to variations in their starting values. It is sometimes illustrated by the observation that a butterfly's wings in one part of the world could mean the difference between snow and sunshine in another part of the world.

## How Is Chaos Theory Applied in Math?

Chaos theory is often used to describe dynamic systems where the number of relevant variables may be much higher than the ability of computational models to account for. One example is in financial markets: Although prices depend on the individual decisions of millions of investors, market behavior can be more predictable when viewed at a mass scale.

## How Is Chaos Theory Used Today?

Chaos theory is used to describe many complicated systems where computational models are limited by the number of unpredictable variables and random factors. For example, weather systems, fluid dynamics, and population cycles can all be described by some elements of chaos theory. Chaos theory is also used in financial markets, because of the complexities of investor behavior.

## What Is the Connection Between Chaos Theory and Fractals?

Fractals are geometric shapes that are self-similar. In other words, a small fragment of a fractal shape may be a mirror image of the entire fractal. This can be compared to natural forces, where simple patterns can create high degrees of complexity. For that reason, fractal geometry can sometimes be used to describe the patterns and repetitions that may occur in chaotic systems.

## The Bottom Line

Chaos theory is a branch of mathematics that deals with disordered or random-seeming mathematical systems. Many of the tools of chaos theory have applications in finance because of the unpredictable behavior of market participants.