Slow acceleration and deacceleration through a Hopf bifurcation: power ramps, target nucleation, and elliptic bursting.

From the periodicity of regional climate change to sustained oscillations in living cells, the transition between stationary and oscillatory behavior is often through a Hopf bifurcation. When a parameter slowly passes or ramps through a Hopf bifurcation there is a delayed transition to sustained oscillations and an associated memory effect where onset is dependent on the initial state of the system. Most theoretical studies of the delay and memory effect assume constant ramp speeds, overlooking the problem of slow parameter acceleration or deacceleration through the Hopf bifurcation. Using both numerical and analytic methods, we show that slow nonlinear ramps can significantly increase or decrease the onset threshold, changing profoundly our understanding of the associated memory effect. We found that slow parameter acceleration increases the threshold, whereas slow deacceleration decreases the threshold. The theory is applied to the formation of pacemakers in the unstirred Belousov-Zhabotinsky reaction and the onset of elliptic bursting in the context of nerve membrane excitability. We show that our results generalize to all systems where slow passage through a Hopf bifurcation is the underlying mechanism for onset.