Self-synchronizing and -desynchronizing systems of oscillators There are biological systems with adaptable frequencies that are able to synchronize their frequencies, mainly individuals (see e.g. reproductive synchrony). In this case, also the phase is typically synchronized. Populations on the other side occasionally desychronize (see e.g. periodical cicadas).
Furthermore, there are lots of physical systems (oscillators) with fixed frequency that are able to synchronize their phases. See e.g. metronome synchronization.
These are my questions:

*

*What are the most simple physical oscillators with adaptable
frequencies that are able to synchronize frequencies by some
interaction?


*What are the most simple physical oscillators with one common fixed frequency that are able to desynchronize phases by some interaction?
 A: A well-known simple model for synchronization in dynamical systems is the Kuramoto–Daido model of coupled oscillators.
A mechanical system that has been shown (e-print) to be equivalent to the Kuramoto-Daido model is two pendulums on top of a free platform, a system which has recently been revisited:

and has been found to exhibit both in-phase and antiphase synchronizations, depending on its parameters, for instance (click to enlarge):

where $b$ measures the coupling strength (it's the ratio between one pendulum mass and the total mass $m/M$) and $r$ quantifies the nonlinearity of the pendulums (and "can be more usefully interpreted as a driving strength in its own right").
As for frequency synchronization, also called "mode locking", it's in general driven by high connectivity or high coupling strength. Probably the sine map is one of the simplest examples (though a toy model, rather than a specific physical system) to exhibit it. This paper (e-print), although old, provides a very readable introduction to the system. For stochastic systems, this paper considers frequency and phase synchronization on the driven noisy harmonic oscillator.
