Codimension-one bifurcation classification conjecture for one-fast, two-slow critical sets

Let gVfg\in V_f define a critical set C[g]\mathcal{C}[g] for a system with one fast variable and two slow variables. Let Dj,k[g,y]\mathcal{D}_{j,k}[g,y] denote the listed degeneracy subsets, with yR2y\in\mathbb{R}^2 a slow-variable value.

Codimension-one bifurcation conjecture. The codimension-one bifurcations of critical sets C[g]\mathcal{C}[g] are characterised by the stated classification: precisely one set Dj,k[g,y]\mathcal{D}_{j,k}[g,y] is non-empty, for precisely one yR2y\in\mathbb{R}^2, and precisely one of the following occurs: a loop or pair of hyperbolae appears in the fold projections at a fold tangency D1,k\mathcal{D}_{1,k}; two cusps annihilate at a cusp tangency D2,k\mathcal{D}_{2,k}; a quadratic fold line folds over to form two cusps in a swallowtail D3,k\mathcal{D}_{3,k}; the projections of two quadratic fold curves become tangent at D4,k\mathcal{D}_{4,k}; the projections of a quadratic fold curve and a cubic cusp intersect at D5,k\mathcal{D}_{5,k}; or the projections of three fold lines intersect at D6,k\mathcal{D}_{6,k}.

This is a proposed classification of codimension-one bifurcations in the one-fast, two-slow case, analogous to the preceding one-fast, one-slow classification. The source provides no evidence that the classification has been proved or refuted.

Sources & referencesView supporting material

Primary source

Karl Nyman, Peter Ashwin and Peter Ditlevsen, “Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems”, arXiv:1902.09203 (2019).

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