Figure 4. Early warning signals for subcritical Hopf bifurcation using the time series generated from the model (equation (2)).

(a) Time series of acoustic pressure depicting the transition from a stable to an alternative stable state. The control parameter k is increased as a function of time, given by k = 1.667 × 10⁻³ t, from 0 to 1. The parameter reaches the Hopf point (k = 0.62) at t = 372. Plot depicting the change in (b) variance and (c) lag-1 autocorrelation as the system approaches the critical transition. We observe increase in variance and autocorrelation well before the transition. We maintained the value of ε such that the non-dimensional noise intensity is 0.02 to match the experimental conditions. Note that the parameter k and the acoustic pressure that we calculate from the model are non-dimensional.