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A simple example of complex behavior is a group of birds in flight, flocking together in a beautiful unfolding pattern. Fascination with complex behavior is as old as civilization, but progress in understanding it theoretically has been slow. From the scientific revolution of the 17th century until recently, scientists assumed that natural systems could be simplified and approached by linear methods. Differential and integral calculus focus in on small segments of curves to reduce them to lines. The dominant analytical paradigm has prevailed because of the availability of methods of linear analysis, not because of any deeply argued conclusions about nature.

Complex systems, however, are nonlinear. Progress in understanding them was made in the 20th century owing to mathematical advances in nonlinear dynamics and to computers ...

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