Pole-movement in oscillators and Energy Balance principle

(1) How do the poles behave in soft-self and hard-self oscillators executing a process of transition from the initial state to the final state and distortion is a desired contamination, on the complex frequency plane? (2) Do the poles ultimately reside on the imaginary axis of the complex frequency plane? (3) If the poles do not reside on the imaginary axis, what is the frequency of oscillation? (4) In such a non-steady non-linear environment where both the frequency and amplitude of oscillation are changing, then how to apply Barkhausen criteria to study oscillation growth? (5) Can we invoke the Energy Balance principle in such situations? These are some of the questions that this paper deals with.