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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:

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

  2. What are the most simple physical oscillators with one common fixed frequency that are able to desynchronize phases by some interaction?

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  • $\begingroup$ What's the meaning of "adaptable frequency" (possibly in mathematical terms)? $\endgroup$
    – Quillo
    Commented Feb 17, 2022 at 17:56
  • $\begingroup$ Would you allow, in your definition of adaptable, something like a (not-simple) pendulum? (In the sense that it does not always oscillate at a specific frequency given some initial condition.) $\endgroup$
    – aghostinthefigures
    Commented Feb 17, 2022 at 17:57
  • $\begingroup$ A pendulum of variable length would qualify. $\endgroup$
    – Hans-Peter Stricker
    Commented Feb 17, 2022 at 18:05
  • $\begingroup$ By "desynchronize phases" you mean they start out with the same phases and evolve to a fixed phase difference? I think this happens, e.g., with two side-by-side pendula oscillating over a free platform. $\endgroup$
    – stafusa
    Commented Feb 17, 2022 at 23:06
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    $\begingroup$ Like those in these figures: here and here. But I found something better and will soon post an (incomplete) answer. $\endgroup$
    – stafusa
    Commented Feb 18, 2022 at 12:09

1 Answer 1

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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:

enter image description here

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

enter image description here

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.

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  • $\begingroup$ Thanks a lot for the answer that gives me a lot of reading. Unfortunately, my question was for systems a) starting with different frequencies and ending up with the same frequency and b) having the same frequency, starting with random (and partially the same) phases and ending up with "maximally different" phases. Maybe I didn't manage to state this clearly enough? (I'll check if your last link answers a.) $\endgroup$
    – Hans-Peter Stricker
    Commented Feb 18, 2022 at 20:08
  • $\begingroup$ @Hans-PeterStricker Take your time. The OP is clear, I guess I didn't make it clear how I'm answering it. The last reference includes oscillators with different frequencies synchronizing via noise. And the 1988 paper is on an oscillator adjusting its frequency to an external forcing — which is conceptually similar, but arguably easier to understand on a first contact, than two oscillators adjusting their frequencies to each other. And I understood that with "maximally different" phases you meant $\pi$, and then that's the "antiphase" the 2020 manuscript describes. $\endgroup$
    – stafusa
    Commented Feb 18, 2022 at 21:21
  • $\begingroup$ This it what I could have made more clearly - and easily so: with "maximally different" I mean "equally distributed over $2\pi$". As an analogy: Consider some $n$ positively charged identical objects that are freely moveable along a ring. Their positions will be equally distributed over $2\pi$. $\endgroup$
    – Hans-Peter Stricker
    Commented Feb 18, 2022 at 23:35
  • $\begingroup$ @Hans-PeterStricker Hmm.. OK. So, for two oscillators antiphase is what you want, but you also want something that generalizes to more oscillators. Is that correct? Then what about waves propagating on a chain of harmonic oscillators coupled with springs? $\endgroup$
    – stafusa
    Commented Feb 19, 2022 at 0:43

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