Math 6410 • ODE

Consider Mathieu's equation for a parametric oscillator:
\[
  \ddot{x}+(\alpha+\beta\cos t)x=0.
\]
\begin{enumerate}
\item[(a)] Suppose that \(\alpha\approx1\), \(\beta\approx0\). Use a perturbation expansion
in \(\beta\) to show that the transition curves for Mathieu's equation are given
approximately by
\[
  \alpha=1-\frac{\beta^2}{12},\qquad
  \alpha=1+\frac{5}{12}\beta^2.
\]
\item[(b)] Now suppose that \(\alpha\approx 1/4+\alpha_1\beta\), \(\beta\approx0\). In the
unstable region near \(\alpha=1/4\), solutions of Mathieu's equation are of the form
\[
  c_1e^{\sigma t}q_1(t)+c_1e^{\sigma t}q_1(t)
\]
where \(\sigma\) is real and positive, and \(q_1,q_2\) are \(4\pi\)-periodic. Derive the
second order equation for \(q_1,q_2\) and perform a power series expansion in \(\beta\)
to show that \(\sigma\approx \pm\beta\sqrt{1/4-\alpha_1^2}\).
\item[(c)] Use part (b) to deduce that solutions of the damped Mathieu equation
\[
  \ddot{x}+\kappa\dot{x}+(\alpha+\beta\cos t)x=0,
\]
where \(\kappa=\kappa_1\beta+O(\beta^2)\), are stable if to first order in \(\beta\),
\[
  \alpha<\frac{1}{4}-\frac{\beta}{2}\sqrt{1-\kappa_1^2}
  \quad\text{or}\quad
  \alpha>\frac{1}{4}+\frac{\beta}{2}\sqrt{1-\kappa_1^2}.
\]
\end{enumerate}