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I set my calculator to radian mode. I enter any random number $x$. Then I repeatedly apply cosine to that number, getting the sequence $$\cos x, \cos(\cos x), \cos(\cos(\cos x)) \ldots$$ and so on. No matter what $x$ I start with, it always converges around $x^* = 0.7390851$. Obviously, $x^*$ is the solution of $\cos x=x$. But why does repeated application of $\cos$ converge around this number?

gt6989b
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Ryder Rude
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1 Answers1

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There are a couple of possibilities:

  1. The sequence will converge.
  2. The sequence will diverge.
  3. The sequence will cycle.

You can check that if $x$ is a fixed point of $f$ and $|f'(x)|<1$ then the fixed point is stable (i.e., the sequence will converge). Validate this for your particular problem. More details can be found at this link.

gt6989b
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  • Can't it chaotically oscillate? (of course not in case of cos(x)) – lesnik Apr 16 '18 at 12:14
  • @lesnik If you mean e.g. like $x/\sin x $ with $x\to \infty$? how is this different from divergence? – gt6989b Apr 16 '18 at 12:15
  • Something like "several values around 2, than several values around 5, than again several values around 2, without some obvious pattern". Something like strange attractor? I do not know the answer, just curious. – lesnik Apr 16 '18 at 12:22
  • @lesnik i've never seen something like this -- in theory possible but in this type of a setting unlikely... not even sure if we can produce such behavior iterating common functions... – gt6989b Apr 16 '18 at 14:28