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Difference systems (described by maps) exhibit much richer dynamical behavior than differential systems, because of the emphasis they place on occasional "high-frequency" transient kicks. Thus, the standard map (with pulsed gravity) displays chaos, while the gravity pendulum does not. Maps also speed up simulations enormously, by summarizing complex dynamics in short form. A new example of richer bahavior, and of dramatic speed up, comes from the representation of interacting electron clouds and ion clouds. Coupled maps are capable of demonstrating the first order phase transitions (from cloud "off" to "on") that are sometimes observed in practice, and enable the extension of electron cloud simulation to include much slower evolving ion clouds.
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