Math 6410 • ODE

The response of a certain biological oscillator, \((x,y)\), \(x\ge 0\), \(y\ge 0\),
to a stimulus of size \(b\) satisfies the differential equation
\[
\begin{aligned}
  \dot{x}&=x-ay+b,\quad &\dot{y}&=x-cy
    && \text{for } x\ge 0,\ y\ge 0;\\
  &&\dot{y}&=-cy
    && \text{for } x=0,
\end{aligned}
\]
with \(a,b,c>0\).
\begin{enumerate}
\item[(a)] Using phase-plane analysis show that there exists a limit cycle, part of which
lies on the \(y\)-axis, when \(c<1\) and \(4a>(1+c)^2\). [Hint: existence of limit cycle
depends on existence of an unstable spiral].

\item[(b)] Show that the period of the orbit is independent of \(b\).
\end{enumerate}