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Physical Mathematics by Kevin Cahill

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18

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Chaos and fractals

18.1 Chaos

Early in the last century, Henri Poincaré studied the three-body problem and found very complicated orbits. In this and other systems, he found that after a transient period, classical motion assumes one of four forms:

1  periodic (a limit cycle),

2  steady or damped or stopped,

3  quasi-periodic (more than one frequency),

4  chaotic.

Example 18.1 (Duffing’s equation) If one attaches a thin piece of iron to the end of a rod that moves sinusoidally in the x direction at frequency ω near two magnets then the x coordinate ...

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