Seemingly chaotic systems like the weather and the financial markets are governed by the laws of chaos theory.

We all have heard about chaos theory, but if you have not or have forgotten what chaos theory is well here you go from interestingengineering.com:

"Chaos theory deals with dynamic systems, which are highly sensitive to initial conditions, making it almost impossible to track the resulting unpredictable behavior. Chaos theory seeks to find patterns in systems that appear random, such as weather, fluid turbulence, and the stock market.

Since the smallest of changes can lead to vastly different outcomes, the long-term behavior of chaotic systems is difficult to predict despite their inherently deterministic nature.

As Edward Lorenz, who first proposed what became commonly known as the Butterfly Effect, eloquently said, "Chaos: When the present determines the future, but the approximate present does not approximately determine the future.""

You may have heard the term about chaos theory as a butterfly flaps its wings in Brazil, and it sets in motion a series of events that eventually lead to a hurricane in Texas. While it might sound implausible that such a tiny act could trigger a massive phenomenon, the essence of what is sometimes described as the butterfly effect goes far beyond this simplistic portrayal. Thus the smallest change can cause a very dynamic outcome in a dynamic system that we live in today.

As many may know that weather forecasts on the same day is a lot more accuracy that one weeks or further out. The reason for this is that changes that are not known, or predicted can cause changes in the weather pattern. This year for example we are supposed to be in a "La Nino" and there is a predicted outcome of this effect. And yet the impact here in the United States and all over are all different. This is due to unpredicted changes in the system that cause a different outcome that was predicted. The Weather professionals are looking at past events such as La Nino over the decades that help them to predict a likely outcome, but it may not be as predicted.

Classic chaotic systems:

Double pendulum example

A classic example of a chaotic system is a double pendulum. This consists of two pendulums connected to each other vertically.

The motion of a double pendulum is incredibly sensitive to initial conditions. A slight difference in the starting position can lead to dramatically different and chaotic patterns. See the attached image showing the chaotic nature of the second pendulum. Note the first pendulum dones not change the pattern (red in example), but the blue is all over the place due to changes in the environment that seem to impact the second greatly from the first.

Lorenz attractor:

Returning to Lorenz, his work on weather modeling led to the elucidation of what is called the Lorenz system. This is a system of ordinary differential equations which have chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system.

The so-called Butterfly Effect originates from the practical consequences of the Lorenz attractor, which means that numerous diverse initial chaotic conditions progress through a complex domain in a manner that lacks repetition, resulting in the overall unpredictability of chaos.

The shape of the Lorenz attractor itself also resembles a butterfly. Upon closer inspection of these systems, a pattern emerges.

Chaotic systems often exhibit fractal patterns within their behavior.

Fractals: Pictures of chaos:

Fractals are geometric shapes or structures that exhibit a unique property: self-similarity. Unlike traditional geometric figures, such as circles or squares, where each part is identical to the whole, fractals possess a fascinating quality—they can be divided into smaller, self-replicating copies of themselves endlessly at various scales.

This has made them an artistic concept as much as a mathematical curiosity.

Apart from their self-similarity feature, fractals also have infinite details. No matter how deeply you delve into the structure, you'll uncover more intricate patterns, mirroring the larger-scale structure. This limitless complexity is a hallmark of fractals.

Fractals in nature:

What is perhaps most interesting about fractals is that they are very commonly seen in nature. From snowflakes and leaves to glacial river channels and mold, fractals appear to be everywhere.

However, the fractals found in nature are approximate fractals, meaning that they display self-similarity over extended but finite scales. Some examples are snowflakes, Romanesco broccoli, and Brefeldia maxima (a type of mold).

In living organisms, fractals are also found in various biological processes. Migrating cells can form fractal patterns through clustering and branching, and nerve cells often exhibit fractal structures due to their surface-to-volume ratio. These fractal patterns play crucial roles in cell physiology and pathology.

In summary, through the lens of chaos theory, we've ventured into the heart of unpredictability, where even the gentlest of actions can set in motion a cascade of unpredictable effects, like the butterfly flapping its wings.

From meteorology and biology to the complexities of financial markets, chaos theory illuminates the hidden structures within apparent chaos. These theories are not mere abstractions but are responsible for shaping our understanding of the world. They challenge our conventional notions of predictability and order, offering a glimpse into the profound complexity that underlies the chaotic aspects of existence.

So what do you think about Chaos Theory and how does this impact you?

Reference: https://interestingengineering.com/science/chaos-theory-fractals-stock-market-weather