Nonoscillation of Mathieu equations with two frequencies

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As is well known, Mathieu’s equation is a representative of mathematical models describing
parametric excitation phenomena. This paper deals with the oscillation problem for
Mathieu’s equation with two frequencies. The ratio of these two frequencies is not necessarily
a rational number. When the ratio is an irrational number, the coefficient of Mathieu’s
equation is is quasi-periodic, but not periodic. For this reason, the basic knowledge
for linear periodic systems such as Floquet theory is not useful. Whether all solutions of
Mathieu’s equation oscillate or not is determined by parameters and frequencies. Our results
provide parametric conditions to guarantee that all solutions are nonoscillatory. The
advantage of the obtained parametric conditions is that it can be easily checked. Parametric
nonoscillation region is drawn to understand these results easily. Finally, several
simulations are carried out to clarify the remaining problems.