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Are there examples of chaotic systems that are predictable and at the same time sensible to initial conditions? or would that violate the notion of sensibility to initial conditions?

Lets imaginge A system that seems to be sensible to initial conditions, behaves chaotically but always end at the same state (not formally proven that all inputs to the system reach the same state). If someone finds a method that predicts what the system will do but is still chaotic, would the representation of sensibility to initial conditions in that system (or any other systems for that matter) be discared then? And are there examples in the literature of this?

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Yes, of course there are. For instance consider a double pendulum: this is famously chaotic. Now add some arbitrarily small frictional losses in the joints of it. It's now easy to show that, eventually, it will end up in one of a small number of configurations (and the only stable one is hanging straight down unless there is 'stiction' I think). But you can make the time it takes to reach that configuration as long as you like, and the trajectory through phase space it takes to reach it would display SDIC.

(I think you could essentially formally prove that such a system does reach a stable state, and what that state is.)

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