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\begin{document}
\centerline{Math 322 lecture for Friday, Week 7}
\bigskip

\centerline{\sc Dependence on parameters, maximal interval}
\medskip

Here we mention a couple of fairly immediate refinements of the fundamental
existence and uniqueness theorem.  Consider our usual initial value problem:
\begin{align}
  x'=f(x)\\
  x(0)=x_0\nonumber
\end{align}
where~$f\colon E\to\R^n$ is continuously differentiable on the open
subset $E\subset\R^n$ and $x_0\in E$.  The first refinement (dependence on
parameters) says that if we deform~$f$ smoothly and move~$x_0$ slightly, then
the solution deforms smoothly.  The second refinement says that the
solution~$x(t)$ to our initial value problem exists on a uniquely determined
maximal interval about~$t=0$.
\medskip

\textbf{Theorem.} (Dependence on parameters.) Let~$E$ be an open subset
of~$\R^{n+m}$ containing the point~$(x_0,\mu_0)$ where~$x_0\in\R^n$
and~$\mu_0\in\R^m$, and assume~$f\in C^1(E)$.  Then there is a
neighborhood\footnote{A {\em neighborhood} of a point is any set that contains an
open set containing the point.}~$N(x_0)\subseteq\R^n$ of~$x_0$, a
neighborhood~$N(\mu_0)\subseteq\R^m$ of~$\mu_0$, and an~$a>0$ such that 
for all~$y\in N(x_0)$ and for all~$\mu\in N(\mu_0)$, the initial value problem
\begin{align*}
  x'&= f(x,\mu)\\
  x(0)&= y
\end{align*}
has a unique solution~$x=x(t,y,\mu)$ with~$x\in C^1(R)$ where~$R:=[-a,a]\times
N(x_0)\times N(\mu_0)$.
\medskip

{\bf Example.} Let~$A\in M_n(\R)$ and~$x_0\in\R^n$.  Then the solution to the
system~$x'=Ax$ with~$x(0)=x_0$ is~$x(t,x_0,A)=e^{At}x_0$, which is a smooth
function of~$t$,~$A$, and~$x_0$.  In this case, $m=\binom{n}{2}$, and we
identify a point~$\mu\in\R^{m}$ with a matrix~$A_{\mu}$ whose entries, read from
left-to-right, top-to-bottom form~$\mu$.  Thus, $f(x,\mu)=A_{\mu}x$.
\medskip

\textbf{Theorem.} Consider our initial value problem with~$f\in C^1(E)$
and initial condition~$x_0$.  There is an interval~$J=(\alpha,\beta)$ 
with~$\alpha,\beta\in\R\cup\left\{ \pm\infty \right\}$
and a solution~$x(t)$
defined for~$t\in J$ such that if~$y(t)$ is any other solution defined on an
interval~$I$, then~$I\subseteq J$ and~$x(t)=y(t)$ on~$I$.  Further,
if~$\beta\in\R$, i.e., if~$\beta\neq\infty$, then given any compact (closed and
bounded) subset~$K\subset E$, then there exists~$t\in J$ such that~$x(t)\notin
K$.
\medskip

The interval~$J$ is called the {\em maximal interval of existence} and is clearly
uniquely determined.
\bigskip

\Needspace{3\baselineskip}
\centerline{\sc Stable manifold theorem}
\medskip

Last lecture, we started investigating the effect of replacing~$f(x)$
with~$Df_{x_0}$ in~(1) at an equilibrium point~$x_0$, i.e., at a point
where~$f(x_0)=0$.  The first theorem we'll consider which makes this comparison
precise is the {\em stable manifold theorem}.  To state the theorem we need 
to formally introduce the {\em flow} of a vector field, and the idea of a {\em
manifold}.
\medskip

{\bf Flow.} For each~$x_0\in E$, let~$I(x_0)$ be the maximal interval of existence of the
solution to~$(1)$ with initial condition~$x_0$.  Then let
\[
  \Omega:=\left\{ (t,x_0)\in \R\times E: t\in I(x_0) \right\}.
\]
For each~$(t,x_0)\in\Omega$, let~$\phi(t,x_0)$ be the solution to (1) with
initial condition~$x_0$ evaluated at time~$t\in I(x_0)$.  This defines a mapping
\[
\phi\colon\Omega\to\R^n
\]
called the {\em flow} of the vector field~$f\colon E\to\R^n$.  For
each~$t\in I(x_0)$ we define 
\[
  \phi_t(x_0):=\phi(t,x_0).
\]

Our text (Section 2.5) establishes the following properties for the flow:
\begin{enumerate}
  \item $\phi_0(x_0)=x_0$
  \item $\phi_s(\phi_t(x_0))=\phi_{s+t}(x_0)$
  \item $\phi_{-t}(\phi_t(x_0))=x_0$
\end{enumerate}
wherever these expressions make sense.
\medskip

{\bf Example.} Consider the case of a linear system, in which~$f(x)=Ax$ for
some $A\in M_n(\R)$.  Here~$E=\R^n$, and for each~$x_0\in\R^n$, the solution is
\[
  \phi_t(x_0)=x(t) = e^{At}x_0,
\]
and the maximal interval of existence is~$I(x_0)=\R$.  So~$\Omega=\R^{n+1}$, and
the above properties for the flow are easily verified in this special case.  For
instance,
\[
  \phi_s(\phi_t(x_0))=e^{As}(e^{At}x_0)=e^{A(s+t)}x_0=\phi_{s+t}(x_0).
\]
\medskip

{\bf Manifolds.}  Roughly speaking, a manifold is a object that can be
constructed from a collection of open subsets
of~$\R^n$ and a set of instructions for gluing these open sets together.  A
quintessential example is given by an ordinary world atlas.  Each page consists
of a flattened out map of a piece of the earth.  There will be pairs of pages
that overlap along boundaries representing the same regions.  The
drawings of features of the earth on these pages implicitly provide instructions
for gluing the pages together.  If the pages where made of moldable putty, then
it would be possible to piece these pages together to make a shape.  One
possible result, among others would be a sphere, and so we say the sphere is
a manifold.  It is two-dimensional since we glue together open subsets
of~$\R^2$ to make it.  We now move on to the formal definition.
\medskip

\textbf{Definition.} A \emph{metric space} is a set~$X$ with a {\em distance
function} or {\em metric},
\[
d\colon X\times X\to\R
\]
that is positive definite, symmetric, and obeys the triangle inequality:
\begin{enumerate}
  \item $d(x,y)\geq0$ with~$d(x,y)=0$ if and only if~$x=y$
  \item $d(x,y)=d(y,x)$
  \item $d(x,y)\leq d(x,z)+d(z,y)$.
\end{enumerate}

Every metric space~$(X,d)$ is a topological space where a subset~$U\subseteq X$
is {\em open} if for each~$u\in U$, there exists~$r>0$ such that the open ball
of radius~$r$ centered at~$u$ is contained in~$U$:
\[
  B(u,r):=\left\{ x\in X:d(u,x)<r \right\}\subseteq U.
\]
\medskip

\textbf{Definition.} Two subsets~$A,B$ of a metric space~$X$ are {\em
homeomorphic} if there exists a continuous bijection~$f\colon A\to B$ with
continuous inverse.  The mapping~$f$ is then called a {\em homeomorphism}
from~$A$ to~$B$.  (More generally, two topological spaces~$U,V$ are homeomorphic
  if there is a continuous bijection~$f\colon U\to V$ with continuous inverse.)
\medskip

\textbf{Definition.} An~$n$-dimensional differentiable manifold is a connected metric
space\footnote{More generally,~$M$ could be a second-countable Hausdorff
toplogical space.}~$M$ and an open covering~$\left\{ U_{\alpha} \right\}$ (so for
  each~$\alpha$ in some index set,~$U_{\alpha}$ is an open subset of~$M$
and~$M=\cup_{\alpha}U_{\alpha}$) such that:
\begin{enumerate}
  \item for all~$\alpha$, there is a homeomorphism
    \[
      h_{\alpha}\colon U_{\alpha}\to V_{\alpha}
    \]
    where~$V_{\alpha}$ is an open subset of~$\R^n$, and
  \item if~$U_{\alpha}\cap U_{\beta}\neq\emptyset$, the mapping
    \[
      h_{\beta}\circ h_{\alpha}^{-1}\colon h_{\alpha}(U_{\alpha}\cap U_{\beta})
      \to h_{\beta}(U_{\alpha}\cap U_{\beta})
    \]
    is continuously differentiable.
\end{enumerate}
Each pair~$(h_{\alpha},U_{\alpha})$ is called a {\em chart}, and the
collection of charts is called an {\em atlas}.  The mapping~$h_{\beta}\cap
h_{\alpha}^{-1}$ are {\em transition functions}.
\medskip

To go back to the rough description we made earlier: each
chart~$(h_{\alpha},U_{\alpha})$ represents a page~$h_{\alpha}(U_{\alpha})$ in the atlas.  The
set~$U_{\alpha}$ is a piece of the manifold (earth), and the
mapping~$h_{\alpha}$ is the rendering of that piece of the earth onto a flat
piece of paper.  On overlaps~$U_{\alpha}\cap U_{\beta}$ on the manifold the corresponding pages
of the atlas have overlaps~$h_{\alpha}(U_{\alpha}\cap U_{\beta})$
and~$h_{\beta}(U_{\alpha}\cap U_{\beta})$.
We can glue these together with the transition function~$h_{\beta}\circ
h_{\alpha}^{-1}$.
\medskip

\textbf{Theorem.} (Stable manifold theorem.)  Let~$E\subseteq\R^n$ and
let~$f\in C^1(E)$.  Suppose that~$f(0)=0$ and that~$Df_0$ has~$k$ eigenvalues
with negative real part and~$n-k$ eigenvalues with positive real part.  Then
there exists a ~$k$-dimensional differentiable manifold~$S$ tangent to the
stable subspace~$E^s$ of the linearized system~$x'=Df_0(x)$ at~$0$ and there
exists an~$(n-k)$-dimensional differentiable manifold~$U$ tangent to the
unstable space~$E^u$ of the linearized system.  Further
\[
  \lim_{t\to\infty}\phi_t(x_0)=0
\]
for any~$x_0\in S$ and
\[
  \lim_{t\to-\infty}\phi(x_0)=0
\]
for any~$x_0\in U$.
\end{document}


