Strange Attraction

The boundary between simplicity and chaos is much narrower than we’d usually like to believe. Teaching first year physics provides many opportunistic to discuss basic situations that are completely “solved,” in the sense that simple, deterministic laws allow us, at least in idealized cases, to perfectly and unequivocally describe all of the dynamics. For example, a simple pendulum, or an planet orbiting its sun. But change the situation just a tiny bit, and all bets are off. For example, the simple pendulum with completely defined motion becomes chaotic when a few repelling magnets are added:

Or a single positive charge in the field of three negative charges:

Capture 4
Starting configuration
Capture 3

What makes this so confounding is that there is no element of “chance” in the strict sense. Everything is still bound by the same deterministic laws. But a tiny change in the initial conditions is enough to create huge, unpredictable changes in the outcome. However, according to Laplace, this is the only kind of chance. That is, if someone rolling a pair of dice could know all of the initial conditions perfectly and be able to calculate the equations of motion rapidly in his head, he would always know what number would turn up each throw. But since even tiny effects are enough to tip the dice from one “final equilibrium” to another, this is impossible even in principle.

Newton’s Method Fractal

This fractal represents the application of Newton’s Method for finding the solution to simple algebraic equations. If more than one solution exists, the method will settle on one of them, depending on where you start looking. You can color the basins of attraction that represent the solution you will eventually land on given each starting condition. Amazingly, the boundary between regions is infinitely complex, meaning you can zoom in as much as you want and still find four solutions sitting adjacent to each other at every interface.

Maybe this explains all of the “soft sciences,” like psychology and economics. Once the system becomes large enough, simple rules give rise to behavior so complex that we cannot rely on formulas as in physics.

Author: lnemzer

Associate Professor Nova Southeastern University

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