Homework 13

For this homework, you can submit up to 3 times.
You need to write the code by yourself.

Q13.1 $$\quad$$ Chains of piecewise linear maps (Example 24.1)

Let us simulate the diffusion process in the chains of piecewise linear maps defined as $\hat{f}(\hat{x}) = \begin{cases} \Lambda \hat{x} & \quad \hat{x} \in [0, 1/2) \\ \Lambda \hat{x} + 1 - \Lambda & \quad \hat{x} \in (1/2, 1] \end{cases}$ extended to the entire real line by $$\hat{f}(\hat{x} + n) = \hat{f}(\hat{x}) + n$$ for $$n \in \mathbb{z}$$. Please write a program to simulate this system. Choose a large number of equally spaced points $$x_i$$ in range [-1, 1] as the initial conditions, and iterate them for a certain number of steps. Since this map is anti-symmetric, we expect the mean drift is zero, so we just record $$x_i^2$$. Figure (a) below is what I got for slope $$\Lambda = 4$$. The y-axis is the mean value of $$x_i^2$$ at each iteration step, and the x-axis is the iteration index. As is shown, the slope of the straight line is approximate 0.499, so the diffusion constant is about 0.2495 in this case.

Set $$\Lambda = 5$$, what is the approximate diffusion constant in your simulation ?

Q13.2 $$\quad$$ modification of Example 24.4

In example 24.4, we choose an appropriate slope of the linear map $$\Lambda = 2(\sqrt{2}+1)$$ such that the critical point is mapped onto the right border of $$\mathcal{M}_{1_+}$$, therefore region $$\mathcal{M}_{2_+}$$ is mapped to $$\mathcal{M}_{0_+}$$ and $$\mathcal{M}_{1_+}$$. Now try to find the appropriate slope $$\Lambda$$ in range [4, 6] such that the critical point is mapped onto the right border of $$\mathcal{M}_{0_+}$$. Now follow the same procedure in example 24.4 to find the dynamical zeta function and try to get the diffusion constant $$D$$ in this case. In order to validate your result, please check $$1/\zeta(0,1) = 0$$. Also, you can simulate diffusion for this particular slope and obtain the numerical diffusion constant to double check your analytical result.

What is the diffusion coefficient in this case ? ( Note, at least 5 significant digits should be accurate. )

Q13.3 $$\quad$$ Diffusion for odd integer $$\Lambda$$ (Exercise 24.1)

Example 24.2 illustrates the process of applying cycle expansion to the piecewise linear map to calculate the diffusion constant for the case $$\Lambda$$ being even. Now let us turn to the case that $$\Lambda$$ is odd. For example, when $$\Lambda = 3$$, interval [0, 1] can be divided into 4 pieces: $$\{[0,1/3), [1/3, 1/2), (1/2, 2/3), [2/3,1]\}$$, with the corresponding symbols $$\{ 0^+, 1^+, 1^-, 0^-\}$$. You can easily verify that $$f(\mathcal{M}_{1^+}) = \mathcal{M}_{0^+} \cup \mathcal{M}_{1^+}$$, so sequences $$1^+1^-$$ and $$1^+0^-$$ are pruned. Similar argument goes for region $$1^-$$. In this sense, we can treat sequences $$\{1^+0^+, 1^-0^-, 1^+1^+0^+, 1^-1^-0^-, \cdots\}$$ as new symbols, and we get the dynamical zeta function as $1/\zeta = 1 - t_{0^+} - t_{0^-} - t_{1^+0^+} - t_{1^-0^-} - t_{1^+1^+0^-} - t_{1^-1^-0^-} - \cdots$ You may consult example 24.4 to help you understand the above process. The difference between $$\Lambda$$ even and $$\Lambda$$ odd case is that symbolic dynamics in the latter case has infinite number of symbols, as opposed to the finite expression in the $$\Lambda$$ even case: $$1/\zeta = 1 - t_{0^+} - t_{0^-} - \cdots - t_{(a-1)^+} - t_{(a-1)^-} \,.$$ After you get the dynamical function, you can calculate the mean cycle length $$\langle n \rangle_\zeta$$ and mean cycle displacement $$\langle n^2 \rangle_\zeta$$ to obtain the diffusion constant.

We can generalize the above analysis to any odd $$\Lambda$$ case. What is the mean cycle length for $$\Lambda=11$$ ?

Q13.4 $$\quad$$ Dependence of diffusion constant on slope -- Q13.1 continued

If the slope $$\Lambda$$ is changed continuously, you may expect that the diffusion constant changes continuously too in a similar way. However, as shown in Chaosbook figure 24.5, the dependence is not a smooth function. This is not the first time we encounter such a situation. Remember that in homework 10, we observe that a small change of parameter in Henon map results in a totally different natural measure. In this sense, structural stability should not be an easy assumption in dynamical systems.

Figure (b) below shows the numerical diffusion constant I got for $$5<\Lambda<6$$. Although only 100 points are chosen in this range, we still can see the non-smoothness of this curve. Try to plot the diffusion constant for range $$3<\Lambda<4$$. What is the approximate maximal diffusion coefficient in this range ?  