Attractors in dynamic systems#

The attactor can be said to be the state of equilibrium that is inherent to a given dynamic system. The dynamic system is described by a mathematical function that receives a set of inputs or readings that capture the state of the system at a given instance in time. Even for a system we might intuitively expect to exhibit random or chaotic behavior over time (the system of virus-spreading through a population, maybe), the corresponding function describing the system can often enough be fairly short and simple. And, given it has a function in the first place, it is a given that however unusual or chaotic the result may appear it cannot in fact be random; it is a deterministic system.

When one iterates on the function -- by taking the function's output points and feeding them back to the function as the inputs for the next iteration -- a shape emerges and fills in progressively after a high quantity of iterations have been run. The shape represents the system's equilibrium in the form of a "basin of attraction" where the results of any iteration will fall -- this is what practitioners refer to as the system's "attractor".

For the dynamic systems in which the equilibrium/basin of attraction manifests in a fractal shape, their attactor is deemed a "Strange Attractor" -- presumably because the fractal shape is a fairly strange/complex shape for the attractor to take compared to the alternatives (in some systems the attractor can be a fixed point or a simple oscillation).

In 1963, MIT mathematician and meteorologist Edward Lorenz arrived at perhaps the most famous strange attactor -- a fractal evoking the shape of a butterfly, as it were -- as he studied the property of convection in the Earth's atmosphere (a dynamic system by anyone's standards) from an information-theory lens. The butterfly shape inspired the exploration of the famed "butterfly effect" in this research, wherein testing demonstrated that inputs altered by something as subtle/minute as a butterfly's wings flapping could dramatically alter the output.

Visual representations of fractals are often wondrous in themselves, of course, so it should be no surprise to find oneself having something of a "music of the cosmos" moment to see a fractal depicting the landscape of a complex dynamic system in the realm of chaos theory.

Partial list of sources:
https://www.wondriumdaily.com/order-in-randomness-fractals-and-chaotic-systems/
https://softologyblog.wordpress.com/2017/03/04/2d-strange-attractors/
https://en.wikipedia.org/wiki/Phase_space
https://www.sciencedirect.com/topics/physics-and-astronomy/strange-attractor
https://www.chaos-math.org/en/chaos-vii-strange-attractors.html
https://www.allthescience.org/what-is-a-strange-attractor.htm
https://uwaterloo.ca/applied-mathematics/future-undergraduates/what-you-can-learn-applied-mathematics/dynamical-systems/strange-attractors