A few months back, while playing around with , I came across something that completely derailed my plans. Strange attractors - fancy math that creates beautiful patterns. At first I thought I'd just render one and move on, but then soon I realized that this is too much fun. When complexity emerges from three simple equations, when you see something chaotic emerge into beautiful, it's hard not to waste some time. I've spent countless hours, maybe more than I'd care to admit, watching these patterns form. I realized there's something deeply satisfying about seeing order emerge from randomness. Let me show you what kept me hooked.
Heads Up Most of what I've learned about strange attractors comes from working on this visualization. If you're seeking advanced mathematical explanations, this might not be for you. My intention here is to share my learnings in an engaging and accessible manner.
The Basics: Dynamical Systems and Chaos Theory
Dynamical Systems are a mathematical way to understand how things change over time. Imagine you have a system, which could be anything from the movement of planets to the growth of a population. In this system, there are rules that determine how it evolves from one moment to the next. These rules tell you what will happen next based on what is happening now. Some examples are, a pendulum, the weather patterns, a flock of birds, the spread of a virus in a population (we are all too familiar with this one), and stock market.
There are two primary things to understand about this system:
Phase Space : This is like a big collection of all the possible states the system can be in. Each state is like a snapshot of the system at a specific time. This is also called the state space or the world state .
: This is like a big collection of all the possible states the system can be in. Each state is like a snapshot of the system at a specific time. This is also called the or the . Dynamics: These are the rules that takes one state of the system and moves it to the next state. It can be represented as a function that transforms the system from now to later.
Figure 1 - Phase Space in Dynamical Systems
For instance, when studying population growth, a phase-space (world-state) might consist of the current population size and the rate of growth or decline at a specific time. The dynamics would then be derived from models of population dynamics, which, considering factors like birth rates, death rates, and carrying capacity of the environment, dictate the changes in population size over time.
Another way of saying this is that the dynamical systems describe how things change over time, in a space of possibilities, governed by a set of rules. Numerous fields such as biology, physics, economics, and applied mathematics, study systems like these, focusing on the specific rules that dictate their evolution. These rules are grounded in relevant theories, such as Newtonian mechanics, fluid dynamics, and mathematics of economics, among others.
... continue reading