Homework 2

You can download the template program for question 4 and 5 from here (right click, "save as").

Q2.1 \(\quad\) A contracting baker’s map (ChaosBook.org version14.5.7, exercise 4.6)

Consider a contracting (or 'dissipative') baker's map, acting on a unit square \([0,1]^2 = [0,1]\times[0,1]\), defined by \[ \begin{aligned} \left( \begin{array}{l} x_{n+1} \\ y_{n+1} \\ \end{array} \right) &= \left( \begin{array}{c} x_n/3 \\ 2y_n \\ \end{array} \right) \quad y_n \leq 1/2 \, , \\ \left( \begin{array}{l} x_{n+1} \\ y_{n+1} \\ \end{array} \right)&= \left( \begin{array}{c} x_n/3 +1/2\\ 2y_n-1 \\ \end{array} \right) \quad y_n > 1/2 \,. \end{aligned} \] This map shrinks strips by a factor of 1/3 in the x-direction, and then it stretches (and folds) them by a factor of 2 in the y-direction. By how much does the state space volume contract for one iteration of the map? (enter numerical value with at least 4 significant figures)

Q1.2 \(\quad\) A limit cycle with analytic Floquet exponent. (ChaosBook.org version14.5.7, exercise 5.1)

There are only two examples of nonlinear flows for which the Floquet multipliers can be evaluated analytically. Both are cheats. One example is the 2-dimensional flow \[ \begin{aligned} \dot{q} &= ~p + q(1-q^2-p^2) \, , \\ \dot{p} &= -q + p(1-q^2-p^2) \, . \end{aligned} \] It is easy to see that this flow has an equilibrium at the origin \((p,\,q) = (0,\,0)\). Is this equilibrium stable or unstable?

Q1.3 \(\quad\) (continuing from Question 2)

Go to polar coordinates \( (q,p) = (r \cos \theta,r \sin \theta) \) and find the limit cycle of this flow. What is the contracting Floquet exponent for this limit cycle?

Q1.4 \(\quad\) Stability of an equilibrium

In this question, you are going to compute stability eigenvalues and eigenvectors of one of the equilibria of the Rössler system.

Start with completing definition of \( \mbox{StabilityMatrix(ssp)} \) in \( \mbox{Rossler.py} \). Replace the 'None' elements of the matrix by the appropriate partial derivatives.

The next task is to call this function from Stability.py. See that in the beginning of Stability.py, we import the Rossler module, where we now have the definition of the stability matrix. Find the line where you are asked to evaluate this matrix at the \( \mbox{eq0} \), and replace its `None' value by the stability matrix at \( \mbox{eq0} \). Functions of other modules in python are called as follows \[ \mbox{ModuleName.FunctionName(arguments)} \] Read through the rest of the code and complete the lines where you see the \( \mbox{#COMPLETE THIS LINE}\) comment. Once you finish, run \( \mbox{Stability.py} \) to see its output.

You should see a flow spiraling out on a plane that is spanned by the real and imaginary parts of the expanding eigenvector. Our goal for showing this to you is to illustrate the "locally linear" behavior of the flow. Here, we pick an initial condition very close to the equilibrium, with a little perturbation in the unstable plane.

When run, Stability.py should also print the stability eigenvalues at the equilibrium in the terminal, type in the real part \( Re \lambda_1 \)of the most expanding eigenvalue in the box below. Please use at least 4 decimal digits in your answer.

Q1.5 \(\quad\) Stability of a periodic orbit

In this problem you are going to calculate the Floquet matrix (the Jacobian for one period) and its eigenvalues and eigenvectors for the shortest periodic orbit of the Rössler system.
In CycleStability.py, we have given you the initial condition and period of this orbit. Remember that the Jacobian satisfies the following differential equation \[ \frac{d}{dt} J^t(x_0) = A(x) J(x_0)\, , x = f^t(x_0)\, , \quad \mbox{initial condition}\, J^t(x_0) = \mathbf{1}\,, \] where \(A(x)\) is the stability matrix. Note that both sides of the above differential equation are \(d \times d\) matrices, and the value of \(A(x)\) depends on where it is evaluated in the state space. In other words, we need to evaluate the above equation along with the orbit. In order to be able to integrate it using our generic integrators, we need to convert this problem into a \(d + d\times d\) dimensional linear ODE, where first \(d\) elements are the state space points, and the remaining \( d \times d \) are the elements of the Jacobian matrix. We have written the velocity function for this extended system in Rossler.py.

Begin this exercise by reading the content of \( \mbox{JacobianVelocity(sspJacobian, t)} \) in \( \mbox{Rossler.py} \) and understand its construction. You are going to integrate this function in CycleStability.py.

Now go to CycleStability.py and complete the line where you specify the initial condition for the Jacobian. Read through the code and complete the line where you need to find the eigenvalues and eigenvectors of the Jacobian.

Run CycleStability.py. You should see the periodic orbit and the Floquet vectors associated with it. Which arrow corresponds to the marginal direction?

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