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In the book of Chaosbook, at the beginning of chapter 6, it is given that

[...]if the attractor is strange, any two trajectories $x(t) = f^t(x_0)$ and $x(t)+\delta x(t) = f^t(x_0 + \delta x_0)$ that start out very close to each other separate exponentially with time [...]

However, we've all seen that the solutions stays in a surface similar to

enter image description here or enter image description here

, so any two solutions can't have an infinite divergence because any two of them stay in a surface similar to given above, and those surfaces will have a finite distance between them, so the solutions cannot diverge infinity apart, so am I taking the given explanation to literal, or am I just missing something because that kind of explanation for the Lyapunov exponent is, kind of, given in almost every chaos lecture, so I'm confused.

Our
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    They don't separate exponentially forever, they only separate exponentially when they are very close to each other. After a while, they are no longer very close to each other, and the separation is no longer exponential. –  Jan 06 '19 at 08:35
  • @Rahul Is there any known time limit that (approximately) how long this characteristic is preserved ? – Our Jan 06 '19 at 08:37
  • As also mentioned in the comment above, the exponential divergence is concerned with the transient behavior; that is, the first stages of separation of two adjacent trajectories. This is also what's used in the definition of the Lyapunov exponents; they are, informally, the averaged exponent of this exponential divergence. After sufficient time has elapsed, the two trajectories reach a maximal distance and do not diverge anymore from one another; that maximal distance is usually called the size of the attractor. – Chris Jan 06 '19 at 09:17
  • @Chris Ok, thanks for your comment, but how long ? I mean, neither for physicists or mathematicians "for a short amount of time" does not have any meaning. You might say that for every such systems, there is some time interval where this exponential divergence characteristic holds, but is there any thing that about the size of that interval ? – Our Jan 06 '19 at 11:07
  • @Rahul Also see my last comment. – Our Jan 06 '19 at 11:10
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    I don't know if there is a general, system-independent answer for that. 1) To derive the Lyapunov exponents you linearize the separation evolution equation; so one answer would be that as long as this linearization holds, to some degree of error, then you're in the exponential divergence regime. 2) If you integrate forward in time two initially neighboring trajectories in the chaotic attractor's basin, you see that they initially diverge exponentially, then sub-exponentially, and their separation finally reaches a plateau; from such a plot you can see the size of the exponential regime. – Chris Jan 06 '19 at 12:02
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    The above are two ways I can think of, in which you can infer the size of the exponential separation regime; hope they help in answering your question! – Chris Jan 06 '19 at 12:05
  • @Chris Thanks, it helped indeed. – Our Jan 06 '19 at 12:06

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