03 May 2020 - Dr. Nithin Nagaraj

Lecture Nos. 15, 16, 17: Introduction to Chaos Theory, 3 requirements for a dynamical system to be chaotic, Topological conjugacy

[Deadline: May 3, 2020]
Instructions: Do not assume anything over and beyond what was defined in the class (unless explicitly asked in the assignment to do so). 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.

  1. Sequence Space. Prove that the cardinality of the sequence space is uncountably infinite. Note: $\Sigma = { (s_0 s_1 s_2 \ldots ) }$ where $s_i = 0$ or $1$ for all $i$.

  2. Distance metric on $\Sigma$. Prove that, $\forall s, t \in \Sigma$, $d(s,t) := \sum_{i=0}^{\infty} \frac{|s_i-ti|}{2^i}$ is a distance metric.

  3. Applying Proximity Theorem. Let $s \in \Sigma$ where $s=(\overline{1011})$. Find a sequence $t \neq s$ and $t \in \Sigma$ such that $d(s,t) < 10^{-5}$. Further, choose your $t$ such that it is a periodic element under the shift map $\sigma$?

  4. Functions. Give an example each for (a) One-to-one function but not onto, (b) Onto function but not one-to-one, (c) One-to-one and onto function.

  5. Dense. Show that the subset of even integers is not dense in the set of Real numbers.

  6. Dense but small measure. Can you create a subset of the reals with arbitrarily small measure $\epsilon > 0$ (length measure) but is still dense in the set of reals? Hint: We know that the set of rationals is dense in the set of reals but its length measure is zero. We want a dense subset but non-zero measure. Devaney’s book has the answer :)

  7. Let $Y$ be the subset of $\Sigma$ such that it contains all sequences which has period length $N=2k$ where $k>0$ is an integer. Is $Y$ a dense subset in $\Sigma$? Justify.

  8. Devaney Chaos - game theoretic version. State the 3 conditions for the dynamical system $(\Sigma, \sigma, d)$ to be chaotic (Devaney’s definition) in terms of a game between Charlie – the challenger, and Preethi – the prover. Be rigorous and explicit in your description.

  9. Transitivity, points with dense or universal orbit. Can you provide a point $s \in \Sigma$ which is a dense orbit (having a Universal orbit) for $(\Sigma, \sigma, d)$? Give an example other than what was provided in class. How many such points with a dense orbit exist for $(\Sigma, \sigma, d)$? (Hint: There are an awful lot of them$!$)

  10. Sensitive dependence on initial conditions. Prove that $(\Sigma, \sigma, d)$ exhibits sensitive dependence on initial conditions with the universal constant $\beta = 1.9$. Is it possible for $(\Sigma, \sigma, d)$ to have a $\beta = 2.0$?

  11. (a) Construct a map where every point is a fixed point.
    (b) Is it possible to construct a map which has only one fixed point and all other points are period-2?
    (c) Construct a map which has three fixed-points and no other periodic points.
    (d) Find all fixed points and periodic points for the map $P(x) = x^2-1$ from $\mathbb{R} \mapsto \mathbb{R}$.

  12. Topological Conjugacy. Using topological conjugacy with the Tent map, find a period-2 point on the Logistic map.