II.On the stability of the solutions of Mathieu's equation

Semi‐discretization method for delayed systems

“…References [9][10][11][12]. The basic stability chart of periodic systems, the Strutt-Ince diagram of Mathieu equation was ÿrst published by van der Pol and Strutt [13].…”

Large scale radial stability density of Hill's equation

“…We note that the boundaries between the stable and unstable zones form a bifurcation diagram, mainly related to subharmonic bifurcations. The first stability diagram known to us is by van der Pol and Strutt [16]. Subsequently, many papers and textbooks have appeared on this subject; we would like to mention Stoker [20], Meixner and Schäfke [15], Levy and Keller [12], Weinstein and Keller [21,22], Pöschel and Trubowitz [17] and Arnold [3,4,5].…”

Delay, Parametric Excitation, and the Nonlinear Dynamics of Cutting Processes

“…This served as a theoretical basis for stability analysis without providing an actual algorithm. This is why the first stability results appeared in the literature only decades later, like the stabilization of the inverted pendulum by vibrating its pivot point vertically at a specific frequency [Stephenson, 1908], or the stability chart of the Mathieu equation [van der Pol & Strutt, 1928;Ince, 1956]. The explanation of how the children's favorite toy, the swing works, was also quite late for the same reason [Levi & Broer, 1995].…”

“…Although, the so-called Hill's infinite determinant method was available in the literature [Hill, 1886;Rayleigh, 1887], van der Pol and Strutt [1928] published the corresponding stability chart (often referred to as Strutt-Ince chart) in analytical form only much later. The chart is shown in Fig.…”