Jianzhong Su et al 2004 Nonlinearity 17 133 doi:10.1088/0951-7715/17/1/009
Jianzhong Su1,4, Jonathan Rubin2 and David Terman3
Show affiliationsRecommended by A I Neishtadt
Elliptic bursting arises from fast–slow systems and involves recurrent alternation between active phases of large amplitude oscillations and silent phases of small amplitude oscillations. This paper is a geometric analysis of elliptic bursting with and without noise. We first prove the existence of elliptic bursting solutions for a class of fast–slow systems without noise by establishing an invariant region for the return map of the solutions. For noisy elliptic bursters, the bursting patterns depend on random variations associated with delayed bifurcations. We provide an exact formulation of the duration of delay and analyse its distribution. The duration of the delay, and consequently the durations of active and silent phases, is shown to be closely related to the logarithm of the amplitude of the noise. The treatment of noisy delayed bifurcation here is a general theory of delayed bifurcation valid for other systems involving delayed bifurcation as well and is a continuation of the rigorous Shishkova–Neishtadt theory on delayed bifurcation or delay of stability loss.
37N25 Dynamical systems in biology (See mainly 92-XX, but also 91-XX)
Issue 1 (January 2004)
Received 31 December 2002, in final form 21 August 2003
Published 6 October 2003
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