07-25-2023-0137 - For small amplitudes and by linearising, the stability of the periodic solution is given by Mathieu's equation (draft)

For small amplitudes and by linearising, the stability of the periodic solution is given by Mathieu's equation:

${\displaystyle \ddot{u}+(a+B\cos t)u=0}$

where $u$ is some perturbation from the periodic solution. Here the $B\cos (t)$ term acts as an ‘energy’ source and is said to parametrically excite the system. The Mathieu equation describes many other physical systems to a sinusoidal parametric excitation such as an LC Circuit where the capacitor plates move sinusoidally. 

https://en.wikipedia.org/wiki/Parametric_oscillator