My dynamical system professor (and the textbooks we use) all claim that the Lyapunov exponent for the Logistic map with $r=4$ ($x_{n+1} = 4x_n(1-x_n)$) is $\log(2)$. Would someone be able to sketch the calculations for me?

  • $\begingroup$ Hello! Please read How do I ask homework questions on Physics Stack Exchange? and edit your question accordingly. Thanks! $\endgroup$
    – Jonas
    Mar 16 at 19:35
  • $\begingroup$ You do know the solution $x_{n}=\sin^{2}(2^{n}\sin^{-1}(\sqrt{x_0}))$, right? $\endgroup$ Mar 16 at 20:41
  • $\begingroup$ I didn't actually. Although i still don't really see how that helps me, the only way we've been calculating the Lyapunov exponent is through the Osedelets theorem ("time" average of the logarithm of map's first derivative). $\endgroup$ Mar 16 at 21:09
  • $\begingroup$ Also are you sure that's correct? from my understanding the case i proposed is chaotic so an analytical solution sounds a bit weird. Edit: i stand corrected there's a solution for r=4, interesting. $\endgroup$ Mar 16 at 21:11
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    $\begingroup$ Due diligence. $\endgroup$ Mar 16 at 21:21

I am careful to not do your homework for you, so I will invite you to "really see" the answer, as per your comment, through the original way (1870) it was elegantly understood by the incomparable Schröder.

Analytically continue the discrete index of the (1870) quasi-periodic solution to your recursion, $$ 𝑥_𝑛=\sin^2(2^𝑛 \arcsin(\sqrt{𝑥_0})), $$ from n (the splinter) to t (the group), a continuous trajectory, $$ 𝑥_t=\sin^2(2^t \arcsin(\sqrt{𝑥_0})), $$ conjugate/equivalent to the tent map, $$ \arcsin(\sqrt{x_t})= 2^t \arcsin(\sqrt{x_0}) ~~~\leadsto \\ \arcsin(\sqrt{x_t})/\arcsin(\sqrt{x_0}) =2^t. $$

  • $\begingroup$ Ah alright i think i understand how to approach it, thank you! $\endgroup$ Mar 17 at 8:58

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