How to quantify amplitude of oscillations in marginally stable systems? Transfer function pole on the Imaginary axis indicates that the system is marginally stable which in time domain can be represented as a sinusoidal motion with constant amplitude and frequency of the Imaginary axis pole. In some applications, oscillations with small amplitude might be acceptable.

Is amplitude of oscillation a (only) function of initial condition of the system?
How to intuitively quantify amplitude of the oscillations for marginally stable system?
Is amplitude of oscillation a (only) function of initial condition of the system?

No.  For a system with a transfer function of the form $H(s) = \frac{\cdots}{s^2 + \omega_0^2}$, the amplitude of oscillation will increase any time it is excited by a signal that has a component at $x(t) = \cos \omega_0 t + \phi$.  Even if you're not exciting it intentionally, random noise will excite that pole.  Basically, the output will be of the form $a(t) \cos \omega_0 t + b(t) \sin \omega_0 t$, where $a(t)$ and $b(t)$ will be Wiener (random-walk) processes.  Such processes have variances that tend to infinity as $t \to \infty$.

How to intuitively quantify amplitude of the oscillations for marginally stable system?

&quot;Big and growing, boss, and I don't think they'll get smaller!&quot;
Possibly by how fast they grow.  But grow they will.

In some applications, oscillations with small amplitude might be acceptable.

If you have a system that's exhibiting persistently small oscillations, then you're seeing the signs of a nonlinear phenomenon called a &quot;limit cycle&quot;.  Basically, there's some oscillator in there that's either inherently small-valued (like the oscillation you may see around the least significant bit of a DAC or ADC), or there's a big oscillation that's mostly not getting to the output.  Either way, if it persists at some amplitude, there's some nonlinear process that's keeping it that way.