Category Archives: Chaos

Sending Your Secrets Safely with Chaos

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The theory of chaos is an extraordinarily broad mathematical topic, and we all have some intuition for what it means when a system is chaotic. The ideas of unpredictability, spontaneity, intractability, turbulence, and perhaps randomness, all come to mind. But deterministic chaos is somewhat different than our intuitions would have us believe. If you watch the shape of a flickering flame, the whitewater in a rocky river, or the price of crude oil in North America, you’re definitely seeing behavior which can’t be described without chaos. But you’re also most likely seeing the effects of any number of random influences on the system, whether a faltering breeze or some oil speculator’s whimsy. Chaos theory deals with the behavior of deterministic systems—that is, systems with no random inputs. All the intricacy and intrigue of chaotic behavior can arise in systems which might seem deceptively uncomplicated, like a pendulum hanging from another pendulum, or three stars orbiting each other.

But if you’ve ever heard of the “butterfly effect” (a term coined by a pioneer of chaos theory, Edward Lorenz), it’s likely your intuition is right about the central feature of deterministic chaos: chaotic systems have high sensitivity to initial conditions.

If chaotic systems are so unpredictable and temperamental, how can we possibly make chaos work for us? One answer is encryption. Continue reading Sending Your Secrets Safely with Chaos


Edward Lorenz’s Strange Attraction

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 [1]. 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