Quasi-Projections of Surfaces with Boundaries

Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems

“…for m = n = 1 whose union contains all codimension one bifurcations, not only due to bifurcation in the fast subsystem. Note that det(D{g, h}) = g x h y − g y h x is the Jacobian of the full system (1) and that the first subset is a local degeneracy Saddle-node:…”

“…These cases can be transformed into the normal forms of Table 4 (these are given in [17]). The cases D 1 1 [g] unfold on varying a typical parameter λ as shown in Fig. 2 a,b,c) and d,e,f) respectively, while D 1 2 [g] unfold as shown in Fig.…”

“…There appears to be connections between global equivalence of critical sets with two distinguished parameters (for m = 1 and n = 2) and the equivalence of vector fields under projection to the slow plane. In particular, several hypothesised degeneracies under strong equivalence also appear as degeneracies of orthogonal projections of vector fields, see for instance [1,9,37,43,47] and references. Crucially, however, such equivalences do not produce degeneracies such as the fold tangency where the manifold structure is lost.…”

“…We furthermore consider only systems (1) with bounded global attractors. We restrict ourselves to vector fields defined on some fixed compact regions M ⊂ R m and N ⊂ R n with open interior and smooth boundaries having outward normals m(x) and n(y) respectively.…”

“…a sequence of continuously parametrized slow segments s k : [0, 1] → M × N . We will assume that either A is a loop entirely within C att [g] or [g] for all θ ∈ (0, 1) • There are trajectories φ(τ ) of the fast system such that α(φ(0)) = s k (1) and ω(φ(0)) = s k+1 (0) for k modulo d.…”