What is chaos?

For you to be reading this blog post right now, for certain things in the present to be true, the events that preceded us must occur in a specific order that any little change in the initial conditions will change everything in the present. But, we all know we cannot take a minute event like “Anne ate oatmeal today instead of cereal” and say that WWIII will not happen because of that. That’s now how the world works. Or, does it?

A popular metaphor for this behavior is that a butterfly flapping its wings in China can cause a hurricane in Texas, also known as the butterfly effect [3].

Chaos theory

Chaos theory is a branch of mathematics focusing on the study of chaos—states of dynamical systems. These dynamical systems are deterministic but appear random and are highly sensitive to initial conditions.

The theory was summarized by Edward Lorenz as:

When the present determines the future, but the approximate present does not approximately determine the future.

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate.It also occurs spontaneously in some systems with artificial components, such as the stock market and road traffic [1].

Before I go into detail the math, let us watch a fun video from Jurassic Park.

Dynamical system

Now, in order to understand chaos theory, we need to understand dynamical systems, which are mathematical models of how quantities evolve over time, which can be recorded by this sequence of states

\[p, f(p), f(f(p)), f(f(f(p))), \dots,\]

where $p$ is some quantity that we want to model.

Example of Dynamical system

For example, let’s say we have a system described by a function

\[f(x) = 2x(1-x),\]

where $x$ can be any between $0\leq x \leq 1$. What happens when you plug in different values of $x$?

When $x = 0.25$, we have

\[\begin{aligned} f(0.25) &= 0.375 \\ f(0.375) &= 0.46875 \\ f(0.46875) &= 0.498047 \\ f(0.498047) &= 0.499992 \\ f(0.499992) &= 0.499999999872. \end{aligned}\]

We notice that at state $x=0.25$, the system tends to 0.5 as time elapses. In fact, the same behavior can be observed for any $x \in [0,1]$.

Example of chaotic system

The function $f(x) = 2x(1-x)$ is an example of a dynamical system, but it is not an example of chaos. Let us make, however, a seemingly innocent modification, switching to the function: \(g(x) = 4x(1 - x).\)

Suddenly, the dynamics get a lot more interesting. For some starting values called fxed points, $g(x)$ will simply get stuck:

\[\begin{aligned} g(0.75) &= 0.75 \\ g(g(0.75)) &= 0.75 \\ g(g(g(0.75))) &= 0.75. \end{aligned}\]

and so on.

It can also get locked in a loop wherein it bounces back and forth between two values forever. For example, using decimal approximations, the starting value

\[x = \frac{1}{8}(5 + \sqrt {5}) \approx 0.90451\]

gives

\[\begin{aligned} g(0.90451) &= 0.3459 \\ g(0.3459) &= 0.90451 \\ g(0.90451) &= 0.3459 \end{aligned}\]

etcetera. A loop of length three is also possible, with $g(x)$ cycling through three different numbers indefinitely. In fact, there are starting values of $x$ for which repeated application of $g(x)$ results in any length of loop.

See for yourself what happens when the seed value $x = 0.9$ is used instead of $x = 0.90451$:

This is the graph of the states. The $x$-axis represents the states/phases; the $y$-axis represents the value. For example, phase 1 would be the value of $g(x)$; phase 2 would be the value of $g(g(x))$ so on and so forth. The orange line is the value from the previous slide. We can see that the graph bounces back and forth between two values. The blue line is the graph of the states when the seed value is 0.9. We notice that the repeating pattern is quickly replaced by something unrecognizable when we reach phase 6. This example demonstrates that even a very small change in the seed value x can lead to a vastly different graph of the dynamical system states. (For those math oriented, the function $g(x)$ is an example of logistic map.)

Example of Lorenz system

Another example of chaotic system is the Lorenz system (Lorenz is often quoted as the father of the field). The Lorenz system is a set of three ordinal differential equations. This system has many solutions in 3D space. The graph here shows the trajectory of different functions with very similar starting values.

From this graph, we can see that even though the trajectories are very different, they always stayed on a double spiral. The points that the trajectories spiral around are called the Lorenz attractors.

Conclusion

Chaos has already had a lasting effect on science, yet there is much still left to be discovered. Many scientists believe that twentieth century science will be known for only three theories: relativity, quantum mechanics, and chaos. Aspects of chaos show up everywhere around the world, from the currents of the ocean and the flow of blood through vessels [2].

Chaos is everywhere around us. And within us.

Thank you.


References

[1] Chaos theory. In: Wikipedia. Page Version ID: 973087636. Aug. 15, 2020. url: <https: //en.wikipedia.org/w/index.php?title=Chaos_theory&oldid=973087636> (visited on 08/18/2020).

[2] Chaos Theory: A Brief Introduction. url: https://courses.seas.harvard.edu/climate/eli/Courses/EPS281r/Sources/Chaos-and-weather-prediction/1-Chaos-Theory-A-Brief-Introduction-IMHO.pdf (visited on 08/18/2020).

[3] The Chaos Theory, Unraveling the Mystery of Life, Samuel Won, TEDxDaculaHighSchool. May 17, 2016. url: https://www.youtube.com/watch?v=Q8JZ4L6Ocic&t=283s (visited on 08/18/2020).