Near a bifurcation point, a dynamical system will undergo what is called critical slowing down -- A loss in resilience to perturbation from equilibrium. See here for more on this phenomenon.
A hallmark of this slowing down is an increase in the variance and lag-1 autocorrelation of a stochastically forced system. As the system approaches a bifurcation point due to slow change in a control parameter, an increase in these statistics is also observed.
A quick argument for this can be made for scalar equations:
Assume there is repeated disturbance in the state variable after each time period $\Delta t$. Between disturbances, the return to equilibrium is approximately exponential, with recovery speed $\lambda$. This can be described as an autoregressive progress :
$$y_{n+1} = \exp(-\lambda \Delta t) y_n + \sigma\epsilon \>.$$
Here, $y_n$ is the state variables deviation from equilirbium, $\epsilon \sim \mathcal{N}(0,1)$, and $\sigma$ is the standard deviation. When written in this way, we see that the autocorrelation, $\alpha$ is equal to $\exp(-\lambda \Delta t$). Assuming that $\lambda$ and $\Delta t$ are independant of $y_n$, then we can observe that the system undergoes critical slowing down, then the recovery speed $-\lambda \rightarrow 0^-$, hence the autocorrelation tends to 1 from below.
My question is: how can I extend this argument to systems of differential equations?