Math 6410 • ODE

Consider a Brusselator system for chemical reactants \(x,y\) given by
\[
\begin{aligned}
  \dot{x} &= 1-4x+x^2y,\\
  \dot{y} &= 3x-x^2y
\end{aligned}
\]
\begin{enumerate}
\item[(a)] Show that the quadrilateral bounded by the lines \(x=0\), \(y=0\), \(y=8+x\) and
\(y=10-x\) is a forward invariant subset. [Hint: need to show that if \(n\) is the outer
normal to a given edge of the quadrilateral, then \(n\cdot(\dot{x},\dot{y})\le0\).]
\item[(b)] Find the fixed points and determine their stability.
\item[(c)] Show that the system has at least one nonconstant periodic solution.
\end{enumerate}