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Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle. (English) Zbl 0783.58053

Summary: We present two hypotheses on the mathematical mechanism underlying bursting dynamics in a class of differential systems: (1) that the transition from continuous firing of spikes to bursting is caused by a crisis which destabilizes a chaotic state of continuous spiking; and (2) that the bursting corresponds to a homoclinicity to this unstable chaotic state. These propositions are supported by a numerical test on the Hindmarsh-Rose model, a prototype of its kind. We conclude by a unified view for three types of complex multi-modal oscillations: homoclinic systems, bursting, and the Pomeau-Manneville intermittency.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
34C23 Bifurcation theory for ordinary differential equations
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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