# Is that a phase transition?

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?

• 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 ? Commented Jul 29, 2023 at 13:42
• Guckenheimer's dynamical systems introduces many types of dynamic transitions. But I have not seen this type specifically. Commented Jul 29, 2023 at 13:44