Differentiability means that if y=f(x) then once you zoom in closely enough, y depends linearly on x. This means that one more digit of y requires precisely one more digit of x.
One more digit of y might require any number more digits of x, depending on the slope.
And what about chaotic systems? The number of digits you require in the initial conditions to get one digit precision in the conditions at time t can grow linearly, and quite rapidly, with t.
One more digit of y might require any number more digits of x, depending on the slope.
And what about chaotic systems? The number of digits you require in the initial conditions to get one digit precision in the conditions at time t can grow linearly, and quite rapidly, with t.