I recently talked about (in https://mathstodon.xyz/@tao/113910070146861518) how solutions in a dynamical systems can be roughly divided into an effective dynamics regime, where simplifying principles such as linearity can be reasonably good approximations, and the more complicated regime of no effective dynamics, in which the behavior can be significantly more nonlinear. For instance, in linear regimes, applying a force in one direction, if followed swiftly by an equal and opposite force in the other direction, will get the state roughly back to where one started; but the same is not true in nonlinear regimes. (If one pulls a spring too far in one direction, one can end up with a broken spring, with no way to return to the initial state, regardless of how one tries to push the spring back in the opposite direction.) (1/7)
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