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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}}\,

The eigenvalues of B\, are \rho _{{1,2}}=\phi \pm {\sqrt  {\phi ^{2}-1}}\,.

Also it is true that \phi ={\frac  {1}{2}}{\mathrm  {Trace}}(B)=\cos {\sqrt  {\delta }}2\pi \,.

So now \rho _{{1,2}}=\cos {\sqrt  {\delta }}2\pi \pm i\sin {\sqrt  {\delta }}2\pi \,.

It is true that -1\leq \phi \leq 1\,.

In case -1<\phi <1\,, the solutions are bounded, oscillatory, and stable.

In case \phi =-1,\,\,\,\cos {\sqrt  {\delta }}2\pi =-1\implies \delta =\left(k+{\frac  {1}{2}}\right)^{2}\,.

In case \phi =1,\,\,\,\cos {\sqrt  {\delta }}2\pi =1\implies \delta =k^{2}\,.


Main Page : Ordinary Differential Equations : Parametric Resonance