Registered: Oct 2003
This is basically an open question, but there are some obvious starting points. The first is to look at what is already well-known behavior in simple rules and how that relates to the harmonic oscillator.
The basic behavior of a 1D oscillator is captured by any cyclic behavior. While the oscillator as a useful model in traditional physics, which shows up quite often, its analog of cyclic behavior is also common in NKS. One sees cyclic behavior, and ultimtely cyclic behavior, in almost any computational exploration of simple rules (meaning one looks at all possibilities in a simple rule system).
Continuing the analogy with the range of possible cyclic behaviors, one can look at cellular automata, where one can find cyclic behavior for any length, say in a reversible CA where one can guarantee these to be irreducible. What one cannot guarantee is any prediction on the size of the periods. With a classical oscillator one can get any period and any length. Discrete systems have natural built-in restrictions on the periods of a given length. The NKS methodology would suggest a search to see what is possible, and answer which rule system is the most flexible in period lengths for ECA's.
Carrying the ECA analogy further, one can ask which of these oscillators are stable in external environments, like those found as local structures with rule 110 (p.677) .
Even more interesting than the 1d oscillator is the situation of coupled oscillators. On p.147, Stephen Wolfram shows how the essential behavior behind some of these oscillators can be captured by simple substitution systems. But he points out in general that there is no quick easy way to go from the traditional trigonometric form of the coupled oscillators to simple rule which predicts its behavior.
That is something not within the realm of the traditional approach to harmonic oscillators.
Finding such rules is a nice problem, which seems like it requires a NKS approach, because known traditional methods are inadequate.
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