Geodesic
Three-Dimensional Domains of Parametric Resonance
Informatics and Automation, Differential equations and dynamical systems, Tome 236 (2002), pp. 304-317

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A linear oscillatory system with many degrees of freedom is considered that has periodic coefficients and depends on three independent parameters, the frequency and amplitude of periodic influence and a parameter of dissipative forces (the last two values are assumed to be small). The instability of the trivial solution (parametric resonance) is examined. General expressions for the domains of simple and combination resonances are obtained. The geometry of the resonance domain is studied, which has the shape of a cone in the three-dimensional parameter space for the most frequent types of periodic influence. As examples, the problems of the stability of elastic beams under periodic loads are considered. 
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     author = {A. P. Seyranian and A. A. Mailybaev},
     title = {Three-Dimensional {Domains} of {Parametric} {Resonance}},
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A. P. Seyranian; A. A. Mailybaev. Three-Dimensional Domains of Parametric Resonance. Informatics and Automation, Differential equations and dynamical systems, Tome 236 (2002), pp. 304-317. http://geodesic.mathdoc.fr/item/TRSPY_2002_236_a30/