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 ramps through a Hopf bifurcation, stability loss is delayed considerably when compared to classical static theory. Inherent to biological, chemical, and physical systems, but often overlooked or misunderstood in the literature are nonlinear ramp problems where a parameter slowly accelerates or de-accelerates through the bifurcation point. In this talk I will show how slow nonlinear ramps can significantly increase or decrease the onset threshold, changing profoundly our understanding of stability loss and delay in dynamic bifurcation problems. I will apply the results to membrane accommodation in nerves, predicting the formation of pacemakers in the Belousov-Zhabotinsky reaction, and predicting the duration of the silent phase in elliptic bursting.
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