What does it take to make a system unpredictable?
If you watch a newspaper blowing in the wind, its flopping will be pretty unpredictable. The forces that govern it are constantly changing, maybe even randomly changing: the breeze fluctuates, the pages flap, the creases spin. As onlookers we would decide that the system is governed by some really complicated rules and forces, and we’d be right. It’s easy to give a system the appearance of randomness when its inputs are (or might as well be) random. But what’s the deeper nature of chaos? In chaotic dynamics, we find that the appearance of randomness, unpredictability, and intricacy of a system’s behavior can arise from a few simple deterministic rules. The strict definition of chaos has been somewhat controversial, but a working standard is:
Chaos is non-periodic behavior in a deterministic system with high sensitivity to initial conditions.
The nature of chaos is that simple rules can give rise to unpredictable and subtle behavior.
We’ll be looking at the Lorenz system, a famous system of differential equations which comes from a 1963 paper Edward Lorenz of MIT published in the Journal of Atmospheric Sciences . The publication was momentous, and the Lorenz system helped spur the development modern chaos theory. In his paper, Lorenz Continue reading Edward Lorenz’s Strange Attraction