Chaos ... is everywhere

STM characterization of an in situ-created stadium-shaped GQD.
Credit: Nature (2024). DOI: 10.1038/s41586-024-08190-6

Chaos is everywhere. From the moon's orbit never following the same exact path to the swing of a pendulum never repeating the same arc, order seemingly issuing forth from perceived disorder resides in the very large/Relativity to the very small/Quantum Mechanics. Within chaos, the term Strange Attractor applies as the swing of the pendulum or the orbit of the moon will stay within a given boundary defined by the attractor. 

Visual representation of a strange attractor. In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve,[2] for a wide variety of starting conditions of the system.

 The definition of chaos, as per Wikipedia. is ...

Chaos theory (or chaology[1]) is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities.[2] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization.

A plot of the Lorenz attractor for values r = 28, σ = 10, b = ⁠8/3 ... AKA The Butterfly Effect

The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions).[4] A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.[5][6]: 181–184 [7]

Why is this important? Well, an intrinsic connect between quantum and relativity through chaos theory may help to develop a viable theory of everything - TOE as chaos operates in similar fashion in both realms, something not proven to exist until now. 

A University of Utah study is shedding light on an important, unsolved physics problem: the relationship between chaos theory – which is based on 300-year-old Newtonian physics – and the modern theory of quantum mechanics.

The evolution of disorder into order by the xenon atoms' nuclear spins is a signature of chaos theory, which, contrary to the popular notion, does not imply complete disorder. Instead, chaos theory describes how weather, certain chemical reactions, planetary orbits, subatomic particles and other dynamic systems change over time, with the changes often highly sensitive to starting conditions.

"When you have a [chaotic] system that is characterized by extreme randomness, it paradoxically can produce ordered behavior after a certain amount of time," says Saam. "There is strong evidence that is happening here in our experiment."

The sensitivity to starting conditions is known popularly as "the butterfly effect," based on the fanciful example that a butterfly flapping its wings in South America might set off subtle atmospheric changes that eventually build into a tornado in Texas.

An animation of a double-rod pendulum at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a vastly different trajectory. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.

Had to insert this graphic yet again as it's a time killer to the max. :)