Chaotic Life

Chaotic about Chaos

PreScript: I am by no means an expert in Chaos Theory and everything here should be taken with a grain of thought and a philosophical outlook.

I have always been fascinated by randomness and how patterns arise out of random conditions. Most of the projects I have had fun with, involve some sort of user input initializing a random variable, which then drives a deterministic process over time Random Art or a random process updating randomly over time - The Blind Watchmaker. Unbeknownst to me, i have been creating chaotic systems with high sensitivity to initial conditions. By definition Chaotic systems are deterministic systems, whose behaviour cannot be accurately predicted with just the initial conditions due to errors in measuring devices. Each measurement can go infinitely smaller than the current precision of the measurement device. "Chaos: When the present determines the future but the approximate present does not approximately determine the future." - Edward Lorenz . This means the predictions we make of the system depend on the accuracy of measurement we have of these initial conditions. We have all heard about the butterfly effect, where a flap of a butterfly can cause a hurricane halfway across the globe, and the underlying theory behind this is chaos. The most common example is a double pendulum, which contains a pendulum attached to the end of another pendulum. The initial positions of the pendulums theta1 and theta2 are extremely sensitive and the system could end up in a wildly different state for extremely tiny changes in angle. Why should anyone study this? There are various scenarios in real world which are inherently chaotic, such as the Stock market and the Weather, two fields where no matter how many measurements we make of the past, there is no certain way to predict the outcome at a specific point and chaos is studied extensively.

Some people I know relate this with Murphy’s law - “Anything that can go wrong, Will go Wrong”. Chaotic systems as a reason for unpredictable and undesirable events happening in our day to day life, for example, the coffee being late leads to spilling it on a stranger and missing the bus. And all of this is at the mercy of a chaotic system. And it is very tantalizing to blame some random event somewhere for an event out of your control. But the same is also true for Positive events, or Yhprum’s law - “Everything that can work, will work.”.

Does that mean we don’t have control over anything and small initial conditions have already decided how things will go? This is a much more deeper and we will have to dive into Block Universe Theory, and how the Future has already happened ( This is for a different time ). But I like to envision this with basins of attractions. To explain basins, let’s take the initial example of the double pendulum with two intial states theta1 and theta2, one for each pendulum. We can take these two angles as axes on a 2d plane. Now the plane denotes all possible initial conditions in which the pendulum could be started, and we can simulate what will happen at every point (x, y) after a certain amount of time has passed. Interestingly, we can see that the systems end up in certain states - Periodic motion where the same positions keep repeating, Neutral position with no energy, or continue exhibiting chaotic behavior with high energy. And if we assign different colors to these states, and plot each point on the graph, we see an interesting pattern start to appear. ( a cool paper to read on this : Qin, Bo, and Ying Zhang. “A novel global perspective: Characterizing the fractal basins of attraction and the level of chaos in a double pendulum.” Chaos, Solitons & Fractals 189 (2024): 115694. ). These are called basins of attraction, where nearby points lead to one of the three final states. This pattern is an infinite fractal based on how zoomed in you are to the boundaries and vary the precision, where we will see the same boundaries appearing over and over.

Basin of Attraction Image Source - Károlyi, György, and Tamás Tel. “New features of doubly transient chaos: complexity of decay.” Journal of Physics: Complexity 2, no. 3 (2021): 035001.

I like to think of the two axes as potential actions you could take and these basins as the outcome in your life at a particular point in time. The three states, Red, Blue and White can be thought of as a Bad outcome, a Good outcome or a Neutral outcome at that specific time point. There are regions on the basin where no matter the initial condition, the outcome will always be Red or in certain regions where the outcome is always Blue. Sure, you never know which state you end up in, but changing your actions a little doesn’t necessarily change the outcome.

The coloring scheme and the outcomes assigned to each color is entirely dependent on the person experiencing it. A specific event which is good for one person might not be ideal for another. Are we deriving a lot of philosophy purely from data? Yes. and are we inferring a lot from these chaotic concepts? Yes. In truth, nature doesn’t care what happens or how an event happens to a person. It merely happens and it’s up to the experiencer to accept and take initiative in changing things. People name this phenomenon differently : chaos, god or universal effectors, serendipity, coincidence. I call this nature and I believe that we have full control on how the future goes - metaphorically, manually adding power to the double pendulum to get where we want.

This post was sparked when one of my professors asked me to define chaos in a discussion about Biological systems ( to be covered in a different post ). I have always been fascinated by fractals and the question sent me on a thought spiral about basins and the philosophy of random events. In conclusion, Things happen chaotically all the time because of random events no one can predict. No event is good or bad by itself. When something goes wrong, don’t beat yourself up thinking how you could have changed this or that. But that is no reason to not try to improve and change the future. So, the next time your cup of coffee is late, stop and take the time to enjoy it instead of trying to run behind the sun.