Abstract
With the use of perturbation expansions and asymptotic matching, bursting in the three-dimensional Bonhoeffer–van der Pol equations is shown to be the result of the interaction of two oscillatory modes, one of small amplitude and the other of large amplitude. The large oscillations are similar to the relaxation oscillations found in the two-dimensional van der Pol system, but the small oscillations have not been fully understood before. This analysis also explains the transition in the two-dimensional system from stable equilibrium to relaxation oscillations: the intermediate stage between the two are the small oscillations.
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