Why is synchronisation only possible for self-sustaining oscillators A self sustained oscillator is any oscillator which obeys the following 3 key properties (Balanov 2009):

*

*They do not damp


*They are capable of oscillating without being driven by an external force.


*The shape, amplitude and time scale of these oscillations are chosen by the oscillating system alone. An outsider cannot easily change them, e.g., by setting different initial condition.
It is not obvious to me why an oscillator that does not follow these properties can not synchronise.
Why is it only these types of oscillators which are capable of synchronisation?
edit:
The above properties define self oscillations as opposed to self oscillators. However the question posed by the title still stands.
 A: There is possibly some confusion in the wording of this question. The properties listed are not a definition for self-oscillators, but for self-oscillations.
After all, at least for synchronization as it's usually understood, these criteria, as the OP wrote them, would be too strict — for instance, you certainly can have synchronization in the presence of damping. Also property 3 seems unnecessary — lots of system which can synchronize have oscillations that depend strongly on the initial conditions.
Property 2 (displaying intrinsic oscillations — which is the title question), however, does apply generally: a system that is not itself oscillatory cannot have its oscillations match (resonate with) another's, by assumption. If it doesn't oscillate by itself, but only when coupled to an oscillating system, then it's being driven by it, not resonating.
Now, a better look at the book referred in the question reveals also a few footnotes and remarks that address the points above. With respect to the first feature (damping):

To be more precise, until the power source lasts, as will be explained below; so they are not perpetuum mobile.
[...] dissipation is ubiquitous [...] The system should simply ﬁnd the way to feed on some source of power in order to compensate for its losses.

And with respect to the 3rd feature (independence from initial conditions):

Except in the case of multistability

For the record, the book's 3 features for self-oscillators are:


*

*they must be non-linear systems

*there must be dissipation in them, and

*there must be a source of power.


Where the need for nonlinearity is explained as

it is an interplay between the non-linear power supply and dissipation that makes self-oscillations possible.

