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--- gibson:teaching:spring-2016:iam950:hw2 [2016/04/05 10:11]
gibson
+++ gibson:teaching:spring-2016:iam950:hw2 [2016/04/05 11:24] (current)
gibson
@@ Line 40: / Line 40: @@
 \begin{eqnarray*}
-\dot{x} &= -y - z \\
+\dot{x} = -y - z \\
-\dot{y} &= x + a y \\
+\dot{y} = x + a y \\
-\dot{z} &= b + z(x-c)
+\dot{z} = b + z(x-c)
 \end{eqnarray*}
-with $a= b = 0.1$ and $c=14$.
+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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