Geodesic
Cauchy problem for the Mathieu equation away from parametric resonance
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 8, pp. 1419-1433

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Four solutions of the Cauchy problem for Mathieu’s equation away from parametric resonance domains are analytically constructed using an asymptotic averaging method in the fourth approximation. Three solutions occur near fractional parameter values at which slow combination phases exist. The fourth solution occurs in the absence of slow phases away from parametric resonance domains and the fractional parameter values. 
@article{ZVMMF_2011_51_8_a5,
     author = {A. F. Kurin},
     title = {Cauchy problem for the {Mathieu} equation away from parametric resonance},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1419--1433},
     publisher = {mathdoc},
     volume = {51},
     number = {8},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_8_a5/}
}
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A. F. Kurin. Cauchy problem for the Mathieu equation away from parametric resonance. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 8, pp. 1419-1433. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_8_a5/