(A) Numerical simulation of equation (S30), see Supplementary ile 1, with μ = 1, b = 1, ω = 2π/78 min-1, q = 0, . Signal x(t) oscillates with a constant amplitude and a period T = 78 min. There is no additive noise and the phase θ(t) increases monotonically with time, θ(t) ~ ω t, so when wrapped in the interval [−π, π] the phase is periodic with T = 78 min. (B-D) Numerical simulations of equation (S30) with μ = 1, b = 1, ω = 2π/78 min-1, q = 0, , and min. Each case B-D shows a different dynamical state that depends on the trajectory of μ(t). In all cases, top panel shows signal x(t) oscillatory behavior showing amplitude fluctuations. Second panel is . Third panel is amplitude of the signal . When μ(t) < 0, the system crosses the Hopf bifurcation and amplitude drops to zero. Fourth panel shows the additive white noise in variable x, . Bottom panel shows phase θ(t) increases monotonically in time θ(t) ~ ω t, when wrapped in the interval [−π, π] the phase is periodic with T = 78 min. Phase is not defined when r = 0.