Analysis of Ringing Oscillator
It can be analyzed for gain/phase response. The theory is defined by the Barkhausen Criteria where the midpoint between;


stable decaying oscillations and   
growing saturated oscillations 
and defined by  unity gain with 180 deg using inverting an additional inverted (180 phase shift) making it positive feedback
,stimulated by any energy in that narrow spectral bandwidth such as an impulse or step pulse after which, it is sustained for a long time.  


This settling time can be measured in cycles and is also defined by a frequency response coefficient called zeta or dampening factor which is inverse to Q the shape of the resonant bandpass filter peak Center f / difference frequency of bandwidth when inputs are attenuated -3dB.

The higher the Q , the more sensitive to changes it becomes.

These are fundamental concepts studied in Electrical Engineering.