For general regularizations of the planar visible fold, the paper proves existence and local uniqueness of a saddle-node bifurcation of limit cycles at the grazing parameter value, with distance scaling ε^{2k/(2k+1)}.
Relaxation oscillations in substrate-depletion oscillators close to the nonsmooth limit
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abstract
In this paper, we describe a novel type of relaxation oscillations occurring in a model of substrate-depletion oscillators. Using geometric singular perturbation theory, with blow-up as a key technical tool, we show that the oscillations in this planar model are produced by a complicated interplay between two stable nodes and a discontinuity set in the singular limit $\varepsilon\rightarrow 0$. This interplay produces a new mechanism for producing relaxation-type oscillations, which we also describe in a more general setting.
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math.DS 1years
2019 1verdicts
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The regularized visible fold revisited
For general regularizations of the planar visible fold, the paper proves existence and local uniqueness of a saddle-node bifurcation of limit cycles at the grazing parameter value, with distance scaling ε^{2k/(2k+1)}.