Blue-sky catastrophe in singularly perturbed systems. (English) Zbl 1092.37030

Shilnikov, Andrey; Shilnikov, Leonid; Turaev, Dmitry

Mosc. Math. J. 5, No. 1, 269-282 (2005).

Summary: We show that the blue-sky catastrophe, which creates a stable periodic orbit of unboundedly increasing length, is a typical phenomenon for singularly perturbed (multi-scale) systems with at least two fast variables. Three distinct mechanisms of this bifurcation are described. We argue that it is behind the transition from periodic spiking to periodic bursting oscillations.

MSC:

37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
34E15	Singular perturbations for ordinary differential equations
58K35	Catastrophe theory
70K70	Systems with slow and fast motions for nonlinear problems in mechanics

Keywords:

saddle-node; global bifurcations; stability boundaries; slow-fast system; bursting oscillations; spikes; excitability