ODEPR1

From Exampleproblems

For the equation $u''+(\delta + \epsilon \cos t + \epsilon \sin 2t) u = 0\,$ show that the conditions for parametric resonance are $\epsilon = 0\,$ and $\delta=k^2\,$ or $(k+1/2)^2, (k=0,1,2,...)\,$ .

Let $\epsilon=0\,$ .

Then $u''+\delta u=0\,$ has two L.I. solutions $\cos \sqrt{\delta}t\,$ and $\frac{\sin\sqrt{\delta}t}{\sqrt{\delta}}\,$ .

So the fundamental matrix is $X(t)=\begin{bmatrix} \cos\sqrt{\delta}t & \frac{\sin\sqrt{\delta}t}{\sqrt{\delta}} \\ -\sqrt{\delta}\sin\sqrt{\delta}t & \cos\sqrt{\delta}t \end{bmatrix}\,$

Notice $X(0)=I\,$ (the identity matrix) so this is a principal fundamental matrix.

The monodromy matrix $B\,$ is the inverse of the fundamental matrix evaluated at zero times the fundamental matrix evaluted at the period of the coefficients.

$B=X^{-1}(0)\cdot X(2\pi) = I\cdot X(2\pi) = X(2\pi) = \begin{bmatrix} \cos\sqrt{\delta}2\pi & \frac{\sin\sqrt{\delta}2\pi}{\sqrt{\delta}} \\ -\sqrt{\delta}\sin\sqrt{\delta}t & \cos\sqrt{\delta}2\pi \end{bmatrix}\,$