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====== Differences ====== This shows you the differences between two versions of the page.

--- gibson:teaching:spring-2016:iam950:hw2 [2016/04/01 10:26]
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+++ gibson:teaching:spring-2016:iam950:hw2 [2016/04/05 11:24] (current)
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 ====== IAM 950 HW2 ======
-You job for HW2 is to compute all periodic orbits of the Lorenz ODE up to period 4 and produce a plot for each, like these.
+For this HW, you have a choice: compute periodic orbits of the Lorenz or the Rössler system. I demoed Lorenz in class, but you will learn a lot doing the numerics for yourself. Or you can do Rössler --it's simpler in a number of ways, but it would be more of an adventure into the unknown.
+----
+** Choice 1: The Lorenz system**
+Compute all periodic orbits of the Lorenz ODE up to period 4 and produce a plot for each, like these.
 {{:gibson:teaching:spring-2016:iam950:period2orbit.png?direct&400|}}{{:gibson:teaching:spring-2016:iam950:period3orbit.png?direct&400|}}
@@ Line 27: / Line 33: @@
 **Step 6:** Compute the periodic orbit, either approximately by adjusting $\eta$ until the trajectory very nearly lands back on its starting point, or better, by setting up a function $(x(t+T), y(t+T), z(t+T)) = \phi(x(t),y(t),z(t),T)$ which maps $x,y,z$ forward time $T$ under the Lorenz dynamics, using numerical integration, and then applying a nonlinear numerical solution method to find a zero of the function $\phi(x,y,z,T) - (x,y,z)$.
+----
+** Choice 2: The Rössler system**
+Compute the first four or five periodic orbits of the Rössler system
+\begin{eqnarray*}
+\dot{x} = -y - z \\
+\dot{y} = x + a y \\
+\dot{z} = b + z(x-c)
+\end{eqnarray*}
+with $a= b = 0.1$ and $c=14$. A long trajectory of the Rössler system looks like this
+{{ :gibson:teaching:spring-2016:iam950:rossler.png?direct&500 |}}
+**Step 1:** Find the equilibria and the eigenvalues of the equilibrium near the origin. What is the
+period of the revolution about the equilibrium and the growth factor per revolution?
+**Step 2:** Let the $x=0$ plane define a Poincare section. Trajectories crossing this plane with $x$ increasing will have $z$ very nearly zero, so the value of $y$ at $x=0$ serves as a good coordinate for a 1d return map. The above picture has a black line drawn from $-22 \leq y \leq -8$ with $x=z=0$. Figure out a good parameterization to $\eta = [0,1]$ of a subset of this line and construct a 1d return map by integrating trajectories from points on it.
+**Step 3:** Approximate the numerical return map from step 2 with an analytic function, then use the fixed points of higher-order iterates of the return map to get initial guesses for periodic orbits.
+**Step 4:** Find periodic orbits numerically by solving a nonlinear equation as described in step 6 for Lorenz.

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