Order in a Chaotic World: Introducing the Chaos Theory

In the early 1960s, meteorologist Edward Lorenz discovered that certain systems are fundamentally unpredictable. His theory unleashes a scientific revolution that bears the name “chaos theory.” It had been stated that simple ground rules can sometimes produce bizarre complexity. Those complex structures often have very elemental sources. This phenomenon and theory is often mentioned in practice, but also in video games such as Life is Strange and films such as Jurassic Park.

So what is the actual definition of chaos theory? What is its connection with mathematics and what does it have to do with predictability and determinism? How can these findings be applied positively in common situations?

The definition

In order to understand chaos theory, it is necessary to discuss how the dictionary describes it:

“Mathematics describing patterns in dynamic systems such as weather, the behaviour of gases and liquids, evolution and so on.” [1]

The chaos theory is, therefore, the mathematics that studies and describes dynamic systems and, as it explains, changing processes over time. In spoken language, chaos means a lack of structure or order. Scientists and mathematicians look at chaos a bit differently. For them, a chaotic world or chaotic issues (issues that are unpredictable, when only a small deviation occurs) of a world can have unimaginable consequences.

Introduction into science

Edward Norton Lorenz was an American mathematician, meteorologist, and professor of meteorology at MIT (Massachusetts Institute of Technology). He studied mathematics at Dartmouth College and Harvard University. He started his career in meteorology in World War II as a weather forecaster for the US Army Air Corps. In an attempt to predict the weather, he got a very different result on his computer model due to only a minimal deviation. This was the starting condition and the base of the “Deterministic Nonperiodic Flow”, which is the standard work for the chaos theory ([2]). [1] [3]

The limits to predictability, demonstrated by Edward, were a shock to many scientists and meteorologists. They thought it would eventually be possible to predict the weather for a longer period of time. [4]

Therefore, the chaos theory states the progress and evolution of changing processes, described with differential equations. Initially, exact solutions were not calculable, but nowadays they can be accessed numerically using the computer. Due to the sensitivity of variations in the initial values ​​of the variable, many of these systems will exhibit a complex and quickly deviating behavior. The predictions made about chaotic systems are very sensitive to small and inevitable (e.g. measurement) errors in the initial state. This ensures that there are limits to how predictable the solutions are. This manifests itself in patterns, which are called “(strange) attractors”.

Attractors and iteration

An attractor is a collection of x-coordinates of points close to an iteration. Take the functions y = x² -1 and y = x:

Taking a value on a parabola, and drawing a horizontal line to y = x from this initial value, the same y-values ​​are retained because the line is drawn horizontally. When you get to the line y = x, you automatically get a new x value that you can enter again in the function y = x² -1.

So an attractor consists of several points: starting- and end-values. The initial values ​​are attracted by the final values. When iterating a function, the attractor can be guessed in a reasonable way by performing the iteration until only a finite number of the same x-values ​​appears.

The link between determinism and predictability

Determinism is a philosophical concept that was derived from science and physics. Its followers believe in orderliness: they think that everything in the universe can be predicted with the help of physical laws. To explain this, a timepiece can be proposed, with a huge number of cogs. Every cog would resemble a law and if one would know all of these, the reality could be seen in its entirety and everything would be predictable. A consequence could be indicated for each cause and vice versa.

Determinism and predictability are closely linked, due to the fact that many scientific models are completely deterministic. This means that if reality can be described at one period of time, the future of that reality can be predicted. Examples are the sun setting today and rising again tomorrow, or the date and place of a solar eclipse and for how long it will be visible. The chaos theory is, therefore, a revolutionary idea that turns determinism upside down, because there are deterministic systems that do not behave according to a linear dynamic. [6]

Fractals

Fractals are figures that visually represent iterative processes, and result in self-uniformity. With an unlimited repetition of the iterative process, a figure appears that shows the same structure on whatever scale. Although the studies of fractals and deterministic chaos initially had a different impetus, they are increasingly linked to each other, because iteration is fundamental in both fractals and deterministic chaos. [5]

A well-known example of a fractal is the Koch curve, or the Koch-Snowflake [6]. The composition of the equilateral triangles is the clearest. With every iteration, an equally large triangle is placed but rotated by 60 degrees. Each side becomes 4 times as big, but 3 times as small. If this process is carried out infinitely, a figure with a finite surface but an infinitely long edge is created.

Examples

There are many examples of the chaos theory that are common and a lot more practical. These are a few well-known ones. [7]

Butterfly effect

The best-known example (or even an alternative name for the phenomenon) of the chaos theory is the butterfly effect. The butterfly effect points out how small deviations in initial states can have enormous and unpredictable effects within a chaotic system (as the chaos theory tells us). One wing stroke of a butterfly in Brazil, for example, can cause a hurricane in Texas just because of a tiny stream of air that can grow into something much bigger. This example symbolizes chaos theory.

In reality, this example is very exaggerated, because one wing beat would not suffice. A lot of fluttering, however, could become a major storm. [8] [9]

Billiard table

Another example of the chaos theory is the billiard table, with 49 colored balls and one white ball.

A computer can calculate the future trajectory of the white ball (although the program must know very precisely the position of all the balls on the table). Supposing you move one of the colored balls just a little, the white ball already takes a different route. A billiard ball that you (seemingly) hit twice in the same way, can continue to roll and hit more balls along the way, causing the two balls to take a different path. The routes of the two balls differ due to the fact that nobody can hit a ball exactly the same (twice!). An immeasurably small difference can ultimately have major consequences. Predictions can be made within such a chaotic system, but even the slightest uncertainty about one of the balls can make those predictions go wild. [9]

Ecosystem

A lesser-known example is an ecosystem. An ecosystem is part of the natural environment. Living aspects, such as animals and plants, and non-living aspects, such as the air, water, and soil, ensure that cycles are continuous. A small change in the living conditions of an organism can have major, and often disastrous, consequences for the entire ecosystem. An ecosystem is therefore also a very complex phenomenon.

Weather

The weather is already a clear example of chaos theory. This idea has been around since Edward Lorentz discovered it when he made a weather model perform calculations. After making the same calculation run twice, there were major differences in the two outcomes.

As already discussed, to know a future state of a system, infinite accuracy is required for all initial conditions. If a meteorologist wants to do a precise weather forecast, all data of every millimeter of the atmosphere must be clear. This is, of course, impossible and not feasible at all. So weather forecasts won’t always be right. [10]

The chaos theory is a very confusing topic and is way more realistic than we think. It’s not only a phenomenon that is issued in video games and movies. Chaotic systems are everywhere around us. Realizing this helps scientist and mathematicians to develop their systems and programs for the better. Consequently, chaos theory also points out how small decisions in our lives can create a major set of different outcomes. With this in mind, we can all decide on the better.

Quoted works

[1]: Schutz, R. (2007–2014). Kernerman Nederlands Leerders dictionary.

[2]: (sd).Projectie van Lorenz Aantrekker. Wikipedia.

[3]: MIT News. (2008, April 16). Edward Lorentz, Father of chaos theory and butterfly effect, dies at 90. From: news.mit.edu: http://news.mit.edu/2008/obit-lorenz-0416

[4]: Groot, C. d. (sd). Chaostheorie: kan een vlinder een orkaan veroorzaken? From: npofocus.nl: https://npofocus.nl/artikel/7959/chaostheorie-kan-een-vlinder-een-orkaan-veroorzaken

[5]: Verhulst, R. (sd). Fractalen en chaostheorie. From: nemokennislink.nl: https://www.nemokennislink.nl/activiteiten/fractalen-en-chaostheorie/

[6]: (sd).The first four iterations of the Koch Snowflake. Wikipedia.

[7]: Roel van Hout, R. R. (2002). Chaostheorie. From: exo.science.ru.nl: http://www.exo.science.ru.nl/bronnen/natuurkunde/chaos.html

[8]: Meijden, J. v. (sd). Het Vlindereffect. Buro Bannink.

[9]: Leys, J. (Regisseur). (2013). Chaos 1: Beweging en Determinisme: Panta Rhei [Film].

[10]: Niemendal, E. (2009, september 7). Chaos-theorie. From: weeronline.nl: https://www.weeronline.nl/chaos-theorie/3073/0