Why are linear systems poor oscillators?
(promoted from comment to answer) 

In order to 'not grow unbound nor damp with time' the characteristics of the system must be 100% right. Nothing in the real world is ideal, so in practice the oscillations will either grow unboundend or damp out.

In practice a system is used that amplifies a little, and a non-linearity that compensates this by attenuating a high level more than a lower level. 

To see this, google 'wien-bridge oscillator', and you will see all sorts of non-linear elements used in the feedback loop: incandescent lamps, diodes, FETs, etc.