## The Missing Theorem in Popular Books on Chaos

August 22, 2008 by attractivechaos

Computer plays an important in simulating dynamic systems. For an ordinary system (as opposed to a chaotic system), the simulated trajectory is always close to the one without errors even if numerical methods are approximate.

A chaotic system is different. It is very sensitive to initial values. Tiny error in initial values will lead to a completely different behaviour from the one developed from the original initial values. Nonetheless, people still use computer to simulate the system. If the simulated trajectory is far from the true one without error, what is the computer simulating? Should we trust such results?

We can, but most of popular books on the chaos theory do not explain the reason. I found the answer in a mathematical textbook. Under small errors, although the trajectory given by the computer can be far from the one developed without any error, there must be a true trajectory that is very close to the simulated trajectory. In other words, the computer is not simulating the trajectory we mean to simulate, but simulating another one that we never know. The simulation is still valid if we want to analyze the property of the whole dynamic system (as opposed to studying one trajectory).

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