Stability domain of the second‐order discrete oscillatory system with parametric excitation

This paper presents a quantitative discussion on the stability of the second‐order periodic difference equation, which characterizes the discrete periodic time‐varying system. the periodic parameter discrete system, which corresponds to Mathieu's equation in the continuous system, is represented by a second‐order linear difference equation with a small parameter ε in the varying term. the parameter ε plays an important role in the determination of the stability. By applying McLachlan's method, the expression for the boundary between the stability and the instability can be determined analytically with regard to the parameters contained in the equation. For the case where the periodic parameter of the difference equation is represented as a sum of two even functions, the boundary between stability and instability is determined. the stability can be analyzed in a similar way for the case where the periodic parameter is represented as a sum of N even functions in Fourier series.