ABSTRACT
There is a huge number of papers and monographs dedicated to the investigation of parametric vibrations and parametric instability [13, 16, 19, 22, 33, 42, 43, 49, 64, 68]. Almost all these works are based on the classical perturbation theory presuming smallness of the modulation depth, and this is an obstacle to obtaining a complete picture of the regions of parametric instability. An exception is provided by the fundamental monographs [43, 61] which describe detailed investigations of the Mathieu equation and the Meissner equation for large modulation depths. There is also a method of infinite determinants [61, 67, 68] which allows one to calculate the boundaries of parametric instability regions (parametric resonances) for arbitrary modulation depths. But this method is fairly cumbersome and its numerical realization requires a lot of preliminary analytical work. Eventually, this method reduces to an algebraic eigenvalue problem. The precision of the results obtained by that method can hardly be estimated. There is also the Floquet theory which, in principle, gives the possibility of reducing systems with periodic coefficients to those with constant coefficients. However, calculations based on the Floquet theory yield tangible results only in the framework of the perturbation theory.
