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====== IAM 950 HW2 ======

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|}}

Use the classic parameter values $\sigma = 10, \beta = 8/3, \rho = 28$. Use whatever software system you like (Matlab, Julia, Mathematica, C, Fortran, whatever). Compute the orbits with these steps. 

**Step 1:** Plot the three unstable equilibria with dots. Compute and plot the stable manifold of the origin and a portion its unstable manifold as shown (magenta for stable manifold, blue and green for stable).  

**Step 2:** Plot the pseudo-Poincare section given by the curve $(x,y,z)$

\begin{eqnarray*}
y &= x - 0.01 x (x-8) (x+8) \\
z &= 13 + 0.188 x^2
\end{eqnarray*}

for $-8 \leq x \leq 8$. This 1d curve lies pretty nearly on the attracting set. (I got it by guessing the right form and setting the constants by trial and error). Parameterize this curve numerically by $\eta$ so that $\eta$ is $\pm 1$ at the end points and zero at the center (simplest strategy: let $\eta = x/8$). 

**Step 3:** Let $\eta_{n+1} = f(\eta_n)$ be the return map of $\eta$ onto itself (hand-wavey, because trajectories starting on the $\eta$ curve don't actually come back exactly to $\eta$). Create a table of 
data of $\eta_n, \eta_{n+1}$ pairs by numerical integration of $(x,y,z)$ points starting on the $\eta$ curve. Make a plot of $f(\eta)$ using this numerical data.

**Step 4:** Fit a function $f(\eta)$ to the the numerical data from above. A piecewise polynomial with three or four terms should do. Plot this $f(\eta)$ on top of your numerical values. 

**Step 5:** Plot $f^n(\eta)$ versus $\eta$ along with the diagonal to get an estimate of the $\eta$ value of the period-n orbit. Then use the corresponding $x,y,z$ values as an initial guess for a periodic orbit. 

**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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