Homework 10

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Q10.1 \(\quad\) Natural measure of Henon map

Chapter 19 covers basic notations concerning time and space average of physical observables, among which, one important concept is "natural measure" (definition 19.15). Natural measure serves as integration kernel for computing spatial average of some observables, but, as stated in this chapter, natural measure is by no way a smooth function, which makes it a hard object in practice.

Let's turn to Henon map again \[ (x_{n+1},y_{n+1}) = (1-ax^2_n+b y_n,x_n) \] to numerically investigate its natural measure by formula (19.16). Here, we set parameter \(a = 1.4, b=0.3\). Please read file 'HenonMap.py' carefully and set 'case = 1' to finish approximating natural measure. Note the major part of this file is implemented for you, and you only need to reproduce figure (c) below.

Figure (c) shows that the natural measure is singular almost at every visited point. Therefore, space average through natural measure seems not to be a good approach. In the next few chapters, we will see how cycle expansion can bypass this problem. The attractor shown in figure (c) is very thin, and I can only distinguish 3 layers. How many layers does your plot clearly show ?

Q10.2 \(\quad\) Natural measure of Henon map -- continued

Let's choose a slight different set of parameters for Henon map as opposed to Q10.1 : \(a=1.39945219, b=0.3 \). Some people may use the same method to calculate natural measure, but actually, there exists a limit cycle (stable periodic orbit) for this set of parameters. No matter where you start, the trajectory will be trapped to it. So the natural measure in this case is totally different from that in Q10.1, even though the change of parameters is very small. We conclude that natural measure is not only a singular function, but is also highly sensitive to parameter change.

set 'case = 2'. First evolve the system for a transient period; then plot the state sequence to see whether it has landed in this limit cycle. What is the period of this limit cycle ? ( answer should be a positive integer )

Q10.3 \(\quad\) Piece-wise invariant measure (Exercise 19.5)

The piece-wise map figure (b) below has maximal value \(1\) at point \(\alpha = \frac{3-\sqrt{5}}{2}\), and the slopes are \(\pm \Lambda\) with \(\Lambda = (\sqrt{5}+1)/2 \). We can use the same method in example 19.1 to get the invariant measure of this map. First, we obtain the Perron-Frobenius matrix, which is [2 x 2] in this case: \[ \mathbf{L} = \begin{pmatrix} * & 1/\Lambda \\ 1/\Lambda & * \\ \end{pmatrix} \] You are supposed to figure out the numbers at position (*) above. The invariant measure is the eigenvector corresponding to the leading eigenvalue of \(\mathbf{L}\).

What is the invariant measure of the left branch of this map ? Here we assume measure is normalized as indicated by formula (19.4). ( Hint: this system is closed, namely, no points will escape under mapping, so the leading eigenvalue should be 1.)

Q10.4 \(\quad\) Escape rate in Logistic map (Exercise 20.2)

Chapter 20 almost talks about one thing: the relationship between the expectation value ( asymptotic time and space average ) and the leading eigenvalue of evolution operator, which is formula (20.11). Also from example 20.4, you learn that the escape rate in an open system is related to the leading eigenvalue of the Perron-Frobenius operator by formula (20.35). Although, this relation is revealed in chapter 20, the method of computing leading eigenvalue of evolution operator is postponed to the next few chapters.

Here we conduct numerical simulation to get the escape rate in the Logistic map: \[ f(x) = Ax(1 - x) \] on the unit interval. The trajectory of a point starting in the unit interval either stays in the interval forever or after some iterations leaves the interval and diverges to minus infinity ( for \(A>4\) ). Now let's estimate numerically the escape rate, the rate of exponential decay of the number of points remaining in the unit interval. The result for \(A = 6\) is shown in fig(a). Y-axis \(\Gamma(N)\) is the ratio of the number of remaining points after iteration \(N\) to the total number of initial points. X-axis is the number of iterations. The slope of this line is about -0.831, so the escape rate is 0.831. You can try to set \( A = 4.5\) and get an approximate escape rate 0.361. Now set \( A = 5.0\), what is the escape rate ? (at least 2 significant digits should be accurate )

You may have noticed that a large number of initial points are needed to obtain a relatively accurate escape rate for this system. That is why we turn to more elegant method in the next few chapters.

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Logistic map map
henon measure