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\begin{document}
\centerline{Math 322 worksheet for Wednesday, Week 8}
\bigskip

Recall the algorithm for creating the stable and unstable manifolds for
an~$n$-dimensional system~$x'=f(x)$ as an equilibrium point~$x_0$ at
which~$Df_{x_0}$ has~$k$ eigenvalues with negative real part and~$n-k$ eigenvalues with
positive real part:
\begin{enumerate}
  \item Preprocessing:
    \begin{enumerate}[(i)]
      \item 
    If~$x_0\neq0$, substitute~$x-x_0$ for~$x$.  Then
    define~$F(x)=f(x)-Jf(0)x$ to write the system as
    \[
      x'=Jf(0)x+F(x).
    \]
  \item Choose~$P\in M_n(\R)$ such that
    \[
      P^{-1}Jf(0)P=
      \left(\begin{array}{cc}
	  A&0\\
	  0&B
      \end{array} \right)
    \]
where~$A$ has~$k$ eigenvalues with negative real part and~$B$ has~$n-k$
eigenvalues with positive real part.  Define
\[
  U(t) := 
  \left(\begin{array}{cc}
      e^{At}&0\\0&0
  \end{array} \right)
  \quad\text{and}\quad 
  V(t) := 
  \left(\begin{array}{cc}
      0&0\\0&e^{Bt}
  \end{array} \right)
\]
so that
\[
\exp\left(\left(\begin{smallmatrix}
      A&0\\0&B
\end{smallmatrix} \right)t\right)=U(t)+V(t).
\]
\item Let~$y=P^{-1}x$ and~$G(y):=P^{-1}F(Py)$ to get the system
  \[
y' = 
  \left(\begin{array}{cc}
      A&0\\0&B
  \end{array} \right)y
  +G(y).
  \]
    \end{enumerate}
\item Apply the method of successive approximations by iterating
\[
  (Tu)(t,a):= U(t)a+\int_{s=0}^tU(t-s)G(u(s,a))\,ds-\int_{s=t}^{\infty}V(t-s)G(u(s,a))\,ds.
\]
where~$a\in\R^n$ starting at~$u^{(0)}(t,a)=0$.
\item A stable manifold is given as the set of points
\[
  \left( a_1,\dots,a_k,u_{k+1}(0,a_1,\dots,a_k,0,\dots,0),\dots, u_{n}(0,a_1,\dots,a_k,0,\dots,0)\right)
\]
as~$(a_1,\dots,a_k)$ varies in a neighborhood of the origin in~$\R^k$.  
\item For the unstable manifold, replace~$t$ by~$-t$ in the original system to
  get
  \[
y' = 
-\left(\begin{array}{cc}
    A&0\\0&B
\end{array} \right)y
-G(y)
\]
Make the swap~$\phi\colon y\mapsto(y_{k+1},\dots,y_n,y_1,\dots,y_k)$ to get the
system
\[
  (\phi(y))' = 
\left(\begin{array}{cc}
    -B&0\\0&-A
\end{array} \right)\phi(y)
-\phi(G(y)),
\]
apply the method of successive approximations, then swap back, applying~$\phi^{-1}$ to the
points in the resulting manifold.
\end{enumerate}
\newpage

{\bf Problem 1.} Apply the algorithm just outlined to the system
\begin{align*}
  x'&= -x-y^2\\
  y'&= y+x^2
\end{align*}
to approximate stable and unstable manifolds at the equilibrium point~$x_0=(0,0)$.
\medskip

{\bf Preprocessing.} 
\vskip0.5in

$\displaystyle Jf(0,0) =\hspace{1.5in} F(x,y)=$
\vskip0.5in

$\displaystyle P = $
\vskip0.5in

$\displaystyle A =\hspace{1.8in} B= $
\vskip0.5in

$\displaystyle U(t) =\hspace{1.5in} V(t)= $
\vskip0.5in

$\displaystyle G(x,y) =$
\vskip0.5in

The transformed system is:
\vskip0.5in

$\displaystyle y'=$
\newpage

{\bf Successive approximations.} 
To approximate a stable manifold, iterate
\[
  (Tu)(t,a)=
  U(t)a+\int_{s=0}^tU(t-s)G(u(s,a))\,ds-\int_{s=t}^{\infty}V(t-s)G(u(s,a))\,ds
\]
where~$a=(a_1,a_2)$ starting at~$u^{(0)}(t,a)=(0,0)$.
\vskip0.5in

$\displaystyle u^{(1)}(t,a)=$
\vskip1.5in

$\displaystyle u^{(2)}(t,a)=$
\newpage

$\displaystyle u^{(3)}(t,a)=$
\vskip5.5in

An approximation of the stable manifold is given by:
\newpage


{\bf Draw your approximation of the stable manifold on the vector field:}
\bigskip

\begin{center}
  \includegraphics[height=5in]{08wed.png}
\end{center}
\bigskip

{\bf Two problems to consider if you have extra time:} 
\bigskip

I. Approximate the unstable manifold: replace~$t$ by~$-t$,
make the swap~$\phi\colon (x,y)\mapsto(y,x)$, and consider the system
system
\[
  \left(\begin{array}{c}
      y'\\x'
\end{array} \right)
  = 
\left(\begin{array}{cc}
    -B&0\\0&-A
\end{array} \right)
\left(\begin{array}{c}
    y\\x
\end{array} \right)
-\phi(G(x,y)).
\]
After the swap, you should get the same system you considered for the stable
manifold but with $x$ and~$y$ swapped. Add the approximation of the unstable
space to your drawing.
\medskip

II. Find the other equilibrium point for our system.  What can you say about
stable and unstable manifolds near this point?  What is the behavior of the flow
near this point?
\end{document}


