Let's consider a network of interacting dynamical systems comprising 2 populations (A and B) where the mean field description of the dynamics of the 2 populations is captured by the following equations:

\begin{align} \frac{dx_A}{dt} &= - x_A + p_{AA} J_{AA} x_A - p_{AB} J_{AB} x_B + g_A\\ \frac{dx_B}{dt} &= - x_B + p_{BA} J_{BA} x_A - p_{BB} J_{BB} x_B + g_B, \end{align}

where the variables $p_{ij}$ follow each their own ODE dynamics, that depend on some parameter $\alpha$ (the exact form of the equations is not relevant for the question).

It turns out, that by changing the value of $\alpha$, of course the dynamics of $x_A$ and $x_B$ change, but not in a qualitative manner to indicate a phase transition.

However what changes qualitatively by changing $\alpha$, is the response of $x_A$ to external perturbations. In fact one can define a critical $\alpha_C$ as a transition point, where the response of the system (and in particular $x_A$) is qualitatively different when $\alpha<\alpha_C$ and $\alpha>\alpha_C$.

Can one say that we have a phase transition here?

(The stability of the system doesn't change)

Are you aware of a similar phenomenon on any known system? Or do you have any relevant literature to recommend for further reading on the topic?

  • $\begingroup$ Could you add details about the "response" to the "perturbations" ? I guess if you are using some additive stochastic noise to perturb it then the SDE system may be associated with a phase transition. Or are you just adding à time-dependent forcing term ? $\endgroup$ Commented Jul 29, 2023 at 13:42
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    $\begingroup$ Guckenheimer's dynamical systems introduces many types of dynamic transitions. But I have not seen this type specifically. $\endgroup$ Commented Jul 29, 2023 at 13:44

1 Answer 1


Here is a guess, rather than a firm answer.

Looking only at the first term in each equation, you basically have two exponentially decaying real variables.

The additional terms in each equation correspond up to some proportionality factors to the difference between your two dynamic variables.

Qualitatively, I expect the effect of the additional term to be some kind of proportional feedback control term : Suppose you initiate your two variables at different numbers. The control term measures the difference and readjusts the variables to push them toward one another. Generally, this type of dynamics shows a transition as a function of the control parameter, here the proportionalality constants. If the control is small, each variables evolves on its own. If the control is large, the two variables converge to each other. In the two cases, the two variables decay exponentially so there is no change in the "dynamic behaviour".


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