Slow Passage Through a Hopf Bifurcation: From Oscillatory to Steady State Solutions

Source abstract

This paper investigates the slow passage through a supercritical Hopf bifurcation from a branch of slowly varying periodic solutions to a branch of slowly varying steady states. This analysis is motivated by a recent numerical study of bursting oscillations in an enzymatic system. It was found that the transition from oscillations to steady states is delayed even if the rate of change of the control parameter is extremely small. The delay due to the slow passage is characterized by determining the amplitude of the oscillations at the bifurcation point. Defining $\varepsilon$ as the rate of change of the bifurcation parameter, it is shown that the amplitude is an $O( \varepsilon^{1/4} )$ quantity as $\varepsilon \to 0$. In addition, a particular class of equations leading to relaxation oscillations is considered. It is assumed that frequency $\omega$ of the oscillations at the Hopf bifurcation can be controlled using a second parameter distinct from the bifurcation parameter. It is then shown that the amplitude of the oscillations at the Hopf bifurcation point increases like $\varepsilon ^{1/4} \omega ^{ - 1}$ as $\omega \to 0,\varepsilon$ small but fixed. Finally, this paper concentrates on a specific model of bursting oscillations appearing in an enzymatic system, and the theory is applied.