Transverse 3-parameter unfoldings of vector fields with 'ears' or 'glasses' separatrix graphs carry the numerical invariant φ = -ln ρ / ln λ, so every such family is structurally unstable.
Bifurcations of the polycycle "tears of the heart": multiple numerical invariants
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abstract
"Tears of the heart" is a hyperbolic polycycle formed by three separatrix connections of two saddles. It is met in generic 3-parameter families of planar vector fields. In [arXiv:1506.06797], it was discovered that generically, the bifurcation of a vector field with "tears of the heart" is structurally unstable. The authors proved that the classification of such bifurcations has a numerical invariant. In this article, we study the bifurcations of "tears of the heart" in more detail, and find out that the classification of such bifurcation may have arbitrarily many numerical invariants.
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New structurally unstable families of planar vector fields
Transverse 3-parameter unfoldings of vector fields with 'ears' or 'glasses' separatrix graphs carry the numerical invariant φ = -ln ρ / ln λ, so every such family is structurally unstable.