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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\i{{\mathrm i}}
\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\Im{\operatorname{Im}}

\begin{document}

\title{Differential Equations}
\author{Homework 8}
\date{Due in class Tuesday, May 15, 2018}
\maketitle

\begin{enumerate}

\item (Verhulst, Exercise 13.7.)  Consider the so-called Brusselator
system, 
\begin{gather*}
  \dot x = a - (b+1) \, x + x^2 \, y \,, \\
  \dot y = b \, x - x^2 \, y
\end{gather*}
in which $x$ and $y$ are concentrations (so $x,y \geq 0$), $a$ and $b$
are positive parameters.  Does the Brusselator admit the possibility
of an Andronov--Hopf bifurcation?

\item (Verhulst, Exercise 13.9.)  We are looking for bifurcation
phenomena in the system
\begin{gather*}
  \dot x = (1+a^2) \, x + (2-6a) \, y + f(x,y) \,, \\
  \dot y = -x -2 \, y + g(x,y) 
\end{gather*}
with parameter $a$; $f$ and $g$ can be developed into Taylor series
near $(0,0)$ starting with quadratic terms.

\begin{enumerate}
\item For which values of $a$ does the Poincar\'e--Lyapunov theorem
guarantee asymptotic stability of $(0,0)$?

\item For which values of $a$ is it possible to have a bifurcation of
$(0,0)$?

\item For which values of $a$ does a center manifold exist?

\item For which values of $a$ is it possible to have an Andronov--Hopf
bifurcation?
\end{enumerate}

\item Fix $\sigma>0$.  Suppose that $f(z)$ is a $1$-periodic function,
analytic on every strip
\[
  S_\rho = \{ z \in \C \colon \lvert \Im z \rvert < \rho \}
\]
for some $\rho>\sigma$.  Then the Fourier coefficients
\[
  f_n = \int_0^1 \e^{2 \pi \i n x} \, f(x) \, \d x
\]
satisfy
\[
  \lvert f_n \rvert
  \leq \e^{-2 \pi \sigma \lvert n \rvert} \, \lVert f \rVert_\sigma
\]
where
\[
  \lVert f \rVert_\sigma = \sup_{z \in S_\sigma} \lvert f(z) \rvert
  \,.
\]

\emph{Hint:} Write $f(z) = g(\e^{2 \pi i z})$ and set $w = \e^{2 \pi i
z}$.  Argue that
\[
  f_n = \frac{1}{2 \pi \i} \oint_{\lvert w \rvert = 1}
  \frac{g(w)}{w^{n+1}} \, \d w \,.
\]
Then use Cauchy's theorem to deform the circle of integration to its
maximal extent.

\item Prove the following statement.  Suppose $h$ is $1$-periodic and
analytic on $S_\sigma$ for some $\sigma>0$.  Set $H(x) = x + h(x)$.
Then for every $\varepsilon>0$ there exists $\delta>0$ such that if
$\lVert h \rVert<\delta$, then $H$ is invertible, $H^{-1}$ is analytic
on $S_{\sigma - \varepsilon}$, and it takes the form
\[
  H^{-1}(x) = x - h(x) + g(x)
\]
where
\[
  \lVert g \rVert_{\sigma - \varepsilon}
  \leq \frac{c}{\varepsilon} \, \lVert h \rVert_\sigma^2
\]
for some constant $c$.

(For notation, please refer to Problem~3.)

\emph{Hint:} Show that $g(x + h(x)) = h(x+h(x)) - h(x)$, then use the
Fundamental Theorem of Calculus and finally a Cauchy estimate for
$h'$.

\end{enumerate}

\end{document}
