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Mathematics Illuminated

The Concepts of Chaos

Most of us learned at an early age how an apple falling from a tree... inspired Isaac Newton to describe how the universe behaves by certain predictable rules. But what about when the universe doesn't behave so... predictably? Can mathematics explain the often unpredictable behavior of the physical world?

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The flapping of a butterfly’s wings over Bermuda causes a rainstorm in Texas. Two sticks start side by side on the surface of a brook, only to follow divergent paths downstream. Both are examples of the phenomenon of chaos, characterized by a widely sensitive dependence of the future on slight changes in a system’s initial conditions. This unit explores the mathematics of chaos, which involves the discovery of structure in what initially appears to be randomness, and imposes limits on predictability.

Unit Goals

  • Chaos is a type of nonlinear behavior characterized by sensitive dependence on initial conditions.
  • Chaotic systems can be deterministic, yet unpredictable.
  • Linear systems are solvable because of superposition.
  • Nonlinear systems are often not solvable in an exact sense.
  • Phase space is a way to find the qualitative behavior of nonlinear systems.
  • Equilibrium points can be stable or unstable.
  • Sensitive dependence is easily seen in certain types of iteration procedures.
  • The logistic map provides an example of a system that bifurcates into chaos in a relatively well-understood way.
  • Space scientists use sensitive dependence to plan minimal-fuel routes through the solar system.





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