Homework 4

For this homework, you can submit up to 3 times.
You can download the template program for question 3 and 4 from here (right click, "save as").

Q4.1 \(\quad\) \(D_3\): symmetries of an equilateral triangle (ChaosBook.org version14.5.7, exercise 9.5 a)

Consider group \(D_3 \approx C_{3v} = \{e, \sigma_{12}, \sigma_{23}, \sigma_{31}, C^{1/3}, C^{2/3} \} \), the symmetry group of an equilateral triangle. The meaning of these operation are as follows, \[\begin{aligned} e &- \mbox{identity} \\ \sigma_{12} &- \mbox{reflection leaving 3 invariant, while switching 1 and 2}. \\ & \sigma_{23},\sigma_{31}\mbox{are similarly defined}\\ C^{1/3} & - \mbox{rotation by} 2\pi/3 \\ C^{2/3} & - \mbox{rotation by} 4\pi/3 \\ \end{aligned}\] which one of the following is not a class of \(D_3\)

Q4.2 \(\quad\) \(D_3\): symmetries of a three-billiard game.

A relative periodic orbit inside the three-billiard game is invariant under symmetries \(\{ C^{1/3}, C^{2/3}\}\). The prime period of this orbit is \(T_p\), what is the period for the corresponding full state space periodic orbit ?

Q4.3 \(\quad\) \(C_2\): symmetry of Lorenz flow

Lorenz system \[ \begin{aligned} \dot{x} & = \sigma (y -x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = xy-bz \\ \end{aligned}\] is equivariant under the action of cyclic group \(C_2 = \{ e, C^{1/2}\}\), where \(C^{1/2}\) is the rotation by \(\pi\) about the z-axis. Invariant polynomials can be used to reduce this discrete symmety by transformation: \((u,v,z) = (x^2-y^2, 2xy, z)\). In this exercise, we implement a post-processing method, which means system is integrated in the full state space, after which the orbit is transformed into the invariant polynomials basis. Please read 'Lorenz.py', and complete symmetry operation 'C2', functions 'velocity', 'stabilityMatrix', 'integrator_with_jacob' and 'reduceSymmtry'.

To validate your code, set 'case = 1', you will see a long ergodic trajectory in the full state space and in symmetry reduced space ( figure 'a' and 'b' at the bottom of this page). Set 'case = 2', a relative periodic orbit will show up.

Now, please set 'case = 3' and finish case 3. You are given the initial condition for a relative periodic orbit, and the period. Integrate the orbit for \( 2T_p\) to obtain Jacobian \(J^{2T_p}(x_0)\), and calculate the Floquet multipliers and Floquet vectors. Make sure you get an marginal multiplier. Enter in the following box the expanding Floquet multiplier (not exponent).

Q4.4 \(\quad\) \(C_2\): symmetry of Lorenz flow: continued

Now set 'case = 4' and finish case 4. You are given the intial condition for the same relative periodic orbit. Try to calculate the Floquet multipliers and Floquet vectors for the prime oribt \(T_p\). We define the Floquet matrix to be \( C^{1/2}\cdot J^{T_p}(x_0)\). Enter the expanding multiplier for this new Floquet matrix. Note, you should get a marginal multipler as well. Think about the relation between the answer of this question and that of the last question.

Q4.5 \(\quad\) \(C_n\): cyclic group by a single element.

Consider the n-fold cyclic group \(C_n =\{e, C_n^1, C_n^2,\cdots, C_n^{n-1}\}\), which represents the symmetry of a horizontal plane under rotation through a vertical axis by angle \(2\pi/n\). It has properties \(C_n^i C_n^j = C_n^{i+j}\) and \(C_n^n = e\). Now suppose a flow has \(C_n\) symmetry: \[ f^t(C_n^1\cdot x) = C_n^1 \cdot f^t(x) \] There is a relative periodic orbit in this flow: \(x(0) = C_n^1\cdot x(T_p)\). Here \(T_p\) is prime period, and it is easy to see that the whole period is \(nT_p\): \(x(0) = x(nT_p)\). After reducing this discrete symmetry, we are left in the fundamental domain, which is \(1/n\) of the full state space. The problem now becomes how to relate the Jacobian Matrix in the full state space and the Jacobian in the reduced space. Follow notations in ChaosBook: \(J^{t}(x(0)) = \partial f^t(x(0)) / \partial x(0)\) denotes the Jacobian from \(x(0)\) to \(x(T_p)\). So \(J^{nT_p}(x(0))\) is the Jacobian associated with the full orbit, and \(J^{T_p}(x(0))\) is the Jacobian associated with the trajectory in the fundamental domain.

What is the relation between \(J^{nT_p}(x(0))\) and \(J^{T_p}(x(0))\) ?

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Lorenz ergodic Lorenz reduced