Nonoscillation of Mathieu equations with two frequencies

摘要

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 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.