Bifurcation is a qualitative measure for a dynamical system changing the system parameter. Does the statistical behavior in the system shows phase transition-like characteristics?
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Yes, similarly to the divergent susceptibility of continuous phase transitions, there are statistical analyses of the systems behavior that can reveal it's approaching a bifurcation.
Typically, structurally stable systems return exponentially to equilibrium states. Close to a bifurcation, the systems is not structurally stable - i.e, a small change in a parameter might cause a qualitative change - and the return to equilibrium is slower. This can be observed directly or through its effect on the statistics of noise-induced fluctuations.
The observation of such early warning signs is not without challenges, though (see, e.g., Which System Variables Carry Robust Early Signs of Upcoming Phase Transition?), and the methods mentioned might miss whole classes of transitions (see, e.g., Early warning signs for saddle-escape transitions in complex networks).
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