25 Jun 2020 - Dr. Nithin Nagaraj

Lecture Nos. 18, 19, 20: Cobweb plots, graphical analysis, stability of periodic points, lyapunov exponents, chaos in the sense of lyapunov, invariant distribution, ergodicity, fractals, applications

[Deadline: May 25, 2020]
Instructions: Do not assume anything over and beyond what was defined in the class unless explicitly asked. Show your calculations in entirety. Justify all your answers with appropriate reasoning and arguments. Try to keep your answers brief, but sufficiently long to convey your ideas. If you want a hint to answer any of the questions, please ask me. There is also a BONUS research activity – reward is world-fame :)

  1. For the skew tent map with an arbitrary skew $0 < p < 1$ determine the invariant probability distribution computationally.

  2. Derive analytically the Lyapunov Exponent $\lambda(p)$ for the ergodic skew tent map with an arbitrary skew $0 < p < 1$. Use the invariant probability distribution from (1) above. Hint: Check out pg. 177 of Chapter 6 of Reviews of Nonlinear Dynamics and Complexity (Ed. Schuster) (see the shared folder).

  3. For the skew tent map, for one example of $p$ $(\neq 0.5)$ compute the Lyapunov Exponent both by the formula derived in (2) above and by using a computer program. Thereby, you can verify that you have got the right answer in (2).

  4. Relate the $\lambda(p)$ of (2) with Shannon entropy of the symbolic sequence (itinerary) of a dense non-periodic transitive orbit on the skew tent map (which means avoid all periodic or eventually periodic orbits). You can take the symbolic sequence with symbol $L$ for $0 \leq x < p$ and $R$ for $p \leq x < 1$ for your chaotic trajectory. You can either argue analytically or computationally for this problem.

  5. Using a computer study the dynamics of $Logi_a(x) = ax(1-x)$ for $a= 3 - 0.05$ and $a=3 + 0.05$. What is the difference between these two cases?

  6. Take the two 1D maps defined on $[0,1]$: $Logi_{2}(x = 2x(1-x)$ and $Logi_{4} = 4x(1-x)$. For both the maps: (a) Compute the Lyapunov number and Lyapunov exponent. (b) Take 10 distinct uniformly spaced intervals on $[0,1]$ and compute the stretching/contraction for each one of them on both the maps. (c) Analyze the symbolic dynamics for both the maps with the partition: $[0, 0.5)$ for $L$ and $[0.5,1]$ for $R$. Based on (a), (b) and (c) argue about chaos (or the lack of it) in these maps. (d) Computationally plot the invariant probability distribution for both the maps.

  7. Repetition codes lie on a fractal. Take the real line interval $[0,1)$ and keep removing the middle $1-2^{-n+1}$th interval for $n>1$ odd integer to yield a Cantor set (after infinite iterations). Compute the box-counting dimension of this Cantor set. Show that this is equal to the rate of the Repetition code $\mathcal{R}_n: { 0 \mapsto 0…0_n, 1 \mapsto 1…1_n}$ (Rate $=\frac{1}{n}$).

  8. Computationally construct two kinds of Barnsley’s Ferns – one that was shown in the class (by switching between 4 affine maps) and another by making an exotic fern by either changing the affine maps or the switching pattern or both! (Be creative) :)

**BONUS Research Activity: Try to invent a new application of chaos and become famous :) :) :)