Homework 6

For this homework, you can submit up to 3 times.
There is no numerical questions in this homework, but we provide one simple python script for Q6.3 for you to play with. Download it form here (right click, "save as").

Q6.1 \(\quad\) Projection into slice

In the text book ( chpater 13, Equation (13.6) ), you have seen that the dynamcis in the slice is related to dynamics in the full state space by \[ \hat{v}(\hat{x}) = v(\hat{x}) - \frac{v(\hat{x})^\top \cdot t'}{t(\hat{x})^\top \cdot t'} t(\hat{x}) \] We can treat the above formula as a projection process: projection from full state space to slice by matrix \[ h(\hat{x}) = I - \frac{t(\hat{x}) t'^\top}{t(\hat{x})^\top \cdot t'} \] Then the relation is just \(\hat{v}(\hat{x}) = h(\hat{x}) v(\hat{x})\). Here, all vectors are column vectors, and \(\top\) denotes the transpose of a column vector giving you a row vector. Putting two matrices/vectors together means the usual matrix product; for instance, \(t(\hat{x}) t'^\top\) is the matrix product of a n-dimensional column vector and a n-dimensional row vector giving you a [n x n] matrix. The dot product above is used just to emphasize that the outcome is a number. Actually, Which of the following is not correct about projection \(h(\hat{x})\) ? Please think about the physical meaning.

Q6.2 \(\quad\) Projection of Jacobian

At week 2, we learn the relation between Jacobian in the full state space and that on the Poincare section ( chapter 4, equation (4.25) ): \[ \hat{J}_{ij} = \left( \delta_{ik} - \frac{v'_i \partial_k U'}{v'\cdot \partial U'} \right) J_{kj} \] This transformation looks similar to \(h(\hat{x})\) in Q6.1, right? This motivates us to think about the relation between Jacobian in the full state space and that in the slice. Suppose flow \(f(x, t)\) has continous symmetry \(g(\phi)\). After choosing a proper slice, we reduce the dynamics into the slice. Choose one point in the full state space \(x_1 = x(0)\), and evolve it for a period \(t\), we get \(x_2 = f(x(0), t)\), denote the Jacobian associated with this period as \(J(x_2, x_1)\). At the same time, the dynamics in the slice also traces out a corresponding trajectory from \(\hat{x}_1\) to \(\hat{x}_2\), with \(x_1 = g(\phi_1)\hat{x}_1\) and \(x_2 = g(\phi_2) \hat{x}_2 \). Denote the Jacobian in the slice assoiated with the same period as \(\hat{J}(\hat{x}_2, \hat{x}_1)\). What is the correct relation between \(J(x_2, x_1)\) and \(\hat{J}(\hat{x}_2, \hat{x}_1)\) ?

Hint: try to think about the physical meaning of the following expressions. Also you can write a simple code using two modes system to verify your statement.

Q6.3 \(\quad\) Full tent map (Chapter 14 Example 14.8, 14.10)

Full tent map \[ f(\gamma) = 1 - 2|\gamma - 1/2|, \quad \gamma \in [0, 1] \] has a simple algorithm to find the initial condition corresponding to a specific itinerary \(S^+\): \[ w_{n+1} = \left\{ \begin{array}{l l} w_n & \quad \text{if} \quad s_{n+1} = 0 \\ 1-w_n & \quad \text{if} \quad s_{n+1} =1 \end{array} \right. \] with \(w_1 = s_1\). Then the map point is \( \gamma(S^+) = 0.w_1 w_2 w_3 \cdots = \Sigma_{n=1}^{\infty} w_n/2^n\). For example, we start from point 0.8 and get a perodic orbit: 0.8, 0.4, 0.8 ..., the corresponding itinerary \(S^+ = 101010\cdots\). Then by this algorithm, \(w_1 = s_1 = 1\), \(w_2 = w_1 = 1\), \(w_3 = 1-w_2 = 0\) and so on, we get \(\gamma(S^+)= 0.11001100\cdots = 0.8\). Bingo ! we recover the initial condition successfully.

So along as you know the itinerary sequence (symbolic dynamics), you can calculate the initial condition. Now, itinerary \(\overline{000001}\) denotes the symbolic dynamics of a periodic orbit with length 6. Here, overline means repeating. What is the corresponding initial condition of this orbit ? (choose the smallest one in the periodic orbit)

Q6.4 \(\quad\) "Golden mean" pruned map (Chapter 14 Exercise 14.6)

The first figure at the bottom of this page shows a symmetric tent map on the unit interval such that its highest point belongs to a 3-cycle. The slopes of the two branches have the same magnitude. Please try to find the slope. What is the critical value of this map, namely, the image of the critical point of this map ?

Q6.5 \(\quad\) "Golden mean" map -- continued

If the symbolic dynamics is such that for \(x < 1/2\) we use symbol 0 and for \( x > 1/2 \) we use symbol 1, which one of the following is not an admissible itinerary of a periodic orbit in this map ?

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golden map