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\begin{document}
\centerline{Math 322 lecture for Friday, Week 8}
\bigskip

\centerline{\sc global stable and unstable manifolds}
\medskip

Let~$E$ be an open subset of~$\R^n$ containing the origin,~$0$, and let~$f\colon
E\to\R^n$ be continuously differentiable. Consider the system of differential
equations~$x'=f(x)$.  Suppose~$0$ is an equilibrium point and that~$Df_{0}$
has~$k$ eigenvalues with negative real part and~$n-k$ eigenvalues with positive
real part.  By the stable manifold theorem, in a neighborhood of~$0$ there exists
a~$k$-dimensional stable manifold~$S$ and an ~$n-k$-dimensional unstable
manifold~$U$.  The manifold~$S$ is tangent at~$0$ to the stable space~$E^s$ for
the linearized system~$x'=Df_{0}(x)$.  Similarly,~$U$ is tangent at~$0$ to the
unstable space~$E^u$ for the linearized system.  Further, if~$\phi_t(x)$ is the
flow for the system, then
\[
  \lim_{t\to\infty}\phi_t(p)=0
\]
for all~$p\in S$ and
\[
  \lim_{t\to-\infty}\phi_t(p)=0
\]
for all~$p\in U$.

Define the {\em global stable and unstable manifolds} at the equilibrium
point~$0$ by
\[
  W^s(0):=\cup_{t\leq 0}\phi_t(S)
\]
and
\[
  W^u(0):=\cup_{t\geq 0}\phi_t(U),
\]
respectively. Here, for any subset~$X\subset E$,
\[
  \phi_t (X):=\left\{ \phi_t(x)\colon x\in X \right\}.
\]
It turns out that these manifolds (i) do not depend on our choice of local stable
and unstable manifolds~$S$ and~$U$, (ii) are invariant under~$\phi_t$,
and~(iii) for all~$p\in W^s(0)$,
\[
  \lim_{t\to\infty}\phi_t(p)=0
\]
and for all~$p\in W^u(0)$,
\[
  \lim_{t\to-\infty}\phi_t(p)=0.
\]
\medskip

{\bf Remark.}  There is also version of the stable manifold theorem that applies
to equilibrium points where the linearization has eigenvalues with real part
equal to zero.  It states that if the linearization has $k$ eigenvalues with
positive real part, $j$ eigenvalues with negative real part, and $m=n-k-j$
eigenvalues with zero real part, then there are manifolds~$W^{s}$, $W^{u}$,
and~$W^{c}$ tangent to stable, unstable, and central spaces, respectively, of
the linearization having dimensions $k$, $j$, and $m$, respectively.   These
spaces are invariant under the flow of the system.  See our text, Section~2.7.
\bigskip

\Needspace{3\baselineskip}
\centerline{\sc Hartman-Grobman theorem}
\medskip

We again consider a system~$x'=f(x)$ as above with equilibrium point~$x_0=0$.
(For an arbitrary equilibrium point~$x_0$, just replace~$x$ by~$x-x_0$.)  We
again assume the linearized system has no eigenvalues with real part equal
to~$0$. These equilibrium points are called {\em hyperbolic equilibrium points}.
Roughly, the Hartman-Grobman theorem says that in a neighborhood of~$x_0$, the
system~$x'=f(x)$ and the linearized system~$x'=Df_{x_0}(x)$ are qualitatively
the same, in a way to be made precise below.
\medskip

\textbf{Theorem.} (Hartman-Grobman) Let~$E$ be an open subset
of~$\R^n$ containing the origin, and let~$f\colon E\to\R^n$ be continuously
differentiable with Jacobian matrix~$Jf$.  Suppose that~$0$ is a hyperbolic
equilibrium point of the system~$x'=f(x)$.  Then there exist open
neighborhoods~$U$ and~$V$ of the origin and a homeomorphism (i.e., a continuous
bijection with continuous inverse)
\[
  H\colon U\to V
\]
with~$H(0)=0$ having the following property: for all~$x_{0}\in U$, there is an
interval~$I\subseteq\R$ containing the origin such that for all~$t\in I$,
\[
  H(\phi_t(x_{0}))=e^{Jf(0)t}H(x_0).
\]
\medskip

The theorem says~$H$ maps trajectories of the system~$x'=f(x)$ to trajectories
of the linearized system~$x'=Jf(0)x$ in a neighborhood of the origin.  (Nonzero
equilibria are handled by translating to the origin, as usual.)  The proof of the
theorem is outlined in Section 2.8 of our text and goes, again, by the method of
successive approximations.
\medskip

{\bf Example.}  Consider the system
\begin{align*}
x'&= -x\\
y'&= y+x^2.
\end{align*}
The origin is a hyperbolic equilibrium point, and the linearized system there is
\begin{align*}
  x'&= -x\\
  y'&= y.
\end{align*}
Our text shows how to apply the method of successive approximations to find the
homeomorphism
\[
  H(x,y) = \left( x,y+\frac{1}{3}x^2 \right).
\]
The effect of the mapping is illustrated below with the stable manifolds in blue
and the unstable manifolds in red:
\begin{center}
  \begin{tikzpicture}
    \node at (0,0) {\includegraphics[height=2.5in]{08fri_nonlinear.png}};
    \node at (10,0) {\includegraphics[height=2.5in]{08fri_linear.png}};
    \draw[->] (3.8,0) to node[above] {$H$} (6,0);
  \end{tikzpicture}
\end{center}

The nonlinear system can be solved using the methods we covered during week five
of the semester, and the solution with initial condition~$(x_0,y_0)$ is
\begin{align*}
  x(t) &= x_0e^{-t}\\[8pt]
  y(t) &= \left(y_0+\frac{1}{3}x_0^2\right)e^{t}-\frac{1}{3}x_0^2e^{-2t}.
\end{align*}
To find the stable manifold, we find the points~$(x_{0},y_{0})$ such that the
solution with that initial points converges to $(0,0)$ as~$t\to\infty$.  For the
unstable manifold, we do the same but with $t\to-\infty$. We find 
\begin{align*}
  W^s(0,0) &=  \left\{ \left( x,-\frac{1}{3}x^2 \right):x\in\R
  \right\}\\[8pt]
  W^u(0,0) &= \left\{ (0,y):y\in\R \right\}.
\end{align*}
The solution to the linearized system is
\begin{align*}
  x(t) &= x_0e^{-t}\\
  y(t) &= y_0e^{t}
\end{align*}
with stable and unstable spaces
\begin{align*}
  E^s&= \left\{ (x,0):x\in\R \right\}\\
  E^u&= \left\{ (0,y):y\in\R \right\}.
\end{align*}
Applying~$H$ to the solution of the nonlinear system gives
\begin{align*}
  H(\phi_t(x,y))
  &= H\left(xe^{-t},\left(y+\frac{1}{3}x^2\right)e^{t}-\frac{1}{3}x^2e^{-2t} \right)\\[10pt]
  &= \left(xe^{-t},\left(y+\frac{1}{3}x^2\right)e^{t} \right)\\[10pt]
  &= \left(\begin{array}{cc}
      e^{-t}&0\\
      0&e^{t}
  \end{array} \right)
  \left(\begin{array}{c}
      xe^{-t}\\
\left(y+\frac{1}{3}x^2\right)e^{t}
  \end{array} \right)\\[10pt]
  &= e^{Jf(0,0)t}H(x,y).
\end{align*}
The stable and unstable manifolds for the nonlinear system are mapped by~$H$ to
the stable and unstable spaces, respectively, for the linear system:
\[
  H\left( x,-\frac{1}{3}x^2 \right)
  =(x, 0),
\]
and
\[
  H(0,y)=(0,y).
\]
\end{document}



