It possesses this subjective element (what we consider to be negligible differences) that seems to undermine its standing as a legitimate mathematical discipline.
I think I see what you’re getting at here, but no, “chaotic” is a mathematical property that systems (of equations) either have or don’t have. The idea behind sensitive dependence on initial conditions is that any difference, no matter how small, will eventually lead to diverging trajectories. Since it will happen for arbitrarily small differences, it will definitely happen for whatever difference exists within our ability to make measurements. But the more precisely you measure, the longer it will take for the trajectories to diverge (which is what faul_sname is referring to).
I think I see what you’re getting at here, but no, “chaotic” is a mathematical property that systems (of equations) either have or don’t have. The idea behind sensitive dependence on initial conditions is that any difference, no matter how small, will eventually lead to diverging trajectories. Since it will happen for arbitrarily small differences, it will definitely happen for whatever difference exists within our ability to make measurements. But the more precisely you measure, the longer it will take for the trajectories to diverge (which is what faul_sname is referring to).