Indiana University  Research & Creative Activity Spring 2002 • Volume XXIV Number 3



fractal
fractal
Misiurewicz Points
courtesy Frances Griffin

No Random Order

Michal Misiurewicz looks for order in chaos. As quixotic as it seems, he finds both order and chaos in systems of one dimension. “I concentrate on the real dynamics in dynamical systems,” says Misiurewicz, professor of mathematics at Indiana University–Purdue University Indianapolis. In mathematical terms, he works with trajectories in intervals on the real line, a line on which each point corresponds to a unique real number. Real numbers are all of the rational and irrational numbers. (Misiurewicz’s IU Bloomington colleagues Eric Bedford and Kevin Pilgrim work in complex dynamics; see “Understanding Chaos.”)

Biology provides a simple example. Suppose you are counting the population of a certain species of insect at the same time each year and recording the numbers in a notebook. If there are not too many this year, the insects will have space to reproduce and survive, and next year there will be more. If there are too many, next year there will be fewer. As the years pass, your growing list of numbers will be a trajectory of a dynamical system in one dimension, changing in time.“This is the simplest possible model of dynamics because there is only one variable—population—and the variable is real. Even in such a simple case, you can have chaos,” Misiurewicz says.

Such simple models are relatively easy to investigate, but the simple case turns up in many other more difficult cases. Knowledge of the simple model can be applied to these bewilderingly complicated cases, such as the weather.

Weather models can actually be reduced to a one-dimensional case with their basic features intact (although a lot of specific information will be lost). The result is a one-dimensional system that is chaotic.

“This explains why you can’t predict the weather, except for a few days at a time,” Misiurewicz says. “You can’t make long-term weather predictions no matter how fast computers get.”

Misiurewicz is widely known for his work on one-dimensional systems. Because of his work, certain points found in the boundaries of the Mandelbrot set are now known as Misiurewicz Points. Misiurewicz’s current research efforts are concentrated in three areas. First, because probability theory can be applied effectively in the presence of chaos, he is looking for cases where probabilistic methods can be used for dynamical systems in one dimension.

He’s also looking for possible order structures in chaotic systems. “For intervals on the real line, if you can find a period of three (after three repetitions you return to the initial numerical value), then you know there are periods of every other value as well, and you know you have a chaotic system,” Misiurewicz explains.

This principle was discovered by the Ukrainian mathematician A. N. Sharkovsky, and Misiurewicz uses it in his work. First he finds some points on the interval that are periodic. Then he looks at all periodic points of the system. It turns out there is some order to them—order in chaos.

Finally, Misiurewicz is determining how much chaos there is in a system—a lot or just a little. This is measured by a quantity called entropy.

Misiurewicz points out that chaos is more a part of all of our lives than we realize: “Normal brain activity is chaotic,” he says. “If the activity of a brain is not chaotic, the brain is dead.” —H. K.

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