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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2001 • Question 2

Question

Math 6410 • ODE Fall 2001 • Question 2

Let \(A\) be a real \(N\times N\) matrix, all of whose eigenvalues have negative real parts. Let \[ h:\mathbb {R}\to \mathbb {R}^N \] be a continuous function which is periodic of period \(T\). Let \[ \begin {aligned} (t,v)&\longmapsto u(t,v),\\ \mathbb {R}\times \mathbb {R}^N&\longrightarrow \mathbb {R}^N \end {aligned} \] denote the unique solution of \[ u'=Au+h(t),\qquad u(0)=v. \tag {3} \] For given \(T>0\), define \[ S_T:\mathbb {R}^N\to \mathbb {R}^N \] by \[ S_T(v)=u(T,v). \]

  1. Use the variation of constants formula (and what you know about stability theory for the linear system \(u'=Au\)) to show that \(S_{nT}\) is a contraction mapping, for some positive integer \(n\), sufficiently large.
  2. Show that \[ S_{nT}=\underbrace {S_T\circ \cdots \circ S_T}_{n}, \] the \(n\)-fold composition of \(S_T\) with itself.
  3. Show that if \(S_{nT}\) has a unique fixed point, then so does \(S_T\).
  4. Apply this to show that if \(h\) is a continuous periodic function of period \(T\), then (3) has a unique \(T\)-periodic solution.

Raw LaTeX

Let \(A\) be a real \(N\times N\) matrix, all of whose eigenvalues have negative real parts.
Let
\[
  h:\R\to\R^N
\]
be a continuous function which is periodic of period \(T\). Let
\[
\begin{aligned}
  (t,v)&\longmapsto u(t,v),\\
  \R\times\R^N&\longrightarrow \R^N
\end{aligned}
\]
denote the unique solution of
\[
  u'=Au+h(t),\qquad u(0)=v.
  \tag{3}
\]
For given \(T>0\), define
\[
  S_T:\R^N\to\R^N
\]
by
\[
  S_T(v)=u(T,v).
\]
\begin{enumerate}
\item Use the variation of constants formula (and what you know about stability theory
for the linear system \(u'=Au\)) to show that \(S_{nT}\) is a contraction mapping, for some
positive integer \(n\), sufficiently large.
\item Show that
\[
  S_{nT}=\underbrace{S_T\circ\cdots\circ S_T}_{n},
\]
the \(n\)-fold composition of \(S_T\) with itself.
\item Show that if \(S_{nT}\) has a unique fixed point, then so does \(S_T\).
\item Apply this to show that if \(h\) is a continuous periodic function of period \(T\),
then (3) has a unique \(T\)-periodic solution.
\end{enumerate}

Topic Tags

Linear Systems and Matrix Exponentials Stability of Linear Systems Topological Degree and Fixed Point Methods Variation of Constants Formula