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Consider two models. Firstly, a set of $n$ variables which satisfy a set of differential equations

$$ \frac{d \mathbf{x}}{d t} = \mathbf{A x} $$ where $\mathbf{x}$ is an $n \times 1$ column vector, and $\mathbf{A}$ is a matrix of coefficients $a_{ij}$, which give the effect of variable $x_i$ on $x_j$ (near equilibrium). See for example Will a Large Complex System be Stable?, R. May, Nature 238, 1972.

Secondly, consider a set of $n$ oscillators which synchronize via pairwise interactions. See for example Synchronization in Complex Networks, A. Arenas et al., Phys. Rep. 438, 2008.

My questions is, what are the conceptual similarities between these two models? Is the attaining of synchrony just another perspective on stability? Are they similar, but essentially just different mathematical problems distinct enough to be studied entirely separately?

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  • $\begingroup$ why do you call the differential equation $dx/dt=Ax$ nonlinear ? $\endgroup$
    – Carlo Beenakker
    Commented Mar 24, 2022 at 15:39
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    $\begingroup$ Synchrony can understood as stability of a possibly time-varying "equilibrium", such as a periodic orbit. I recommend Nonlinear Dynamics book by Strogatz $\endgroup$
    – Piyush Grover
    Commented Mar 24, 2022 at 15:43
  • $\begingroup$ Apologies, I was meant to add a point that the equations model things near an equilibrium point. $\endgroup$
    – apg
    Commented Mar 24, 2022 at 23:49

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