Geodesic
Spectral stability criterion and the Cauchy problem for the Hill equation at parametric resonance
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 3, pp. 498-511

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An analytical solution to the Cauchy problem for the Hill equation is constructed by the second-order averaging method for three instability domains, stability domains near the boundaries with the instability domains, and on the boundaries themselves. An unstable exponentially decaying solution is found in the instability domains. A simple (convenient for applications) stability criterion for the trivial solution is formulated in the form of an inequality expressed in terms of the constant component, the amplitudes, and the frequencies of harmonics in the spectrum of the periodic coefficient of the Hill equation. 
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     author = {A. F. Kurin},
     title = {Spectral stability criterion and the {Cauchy} problem for the {Hill} equation at parametric resonance},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {498--511},
     publisher = {mathdoc},
     volume = {49},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_3_a9/}
}
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A. F. Kurin. Spectral stability criterion and the Cauchy problem for the Hill equation at parametric resonance. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 3, pp. 498-511. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_3_a9/