New structurally unstable families of planar vector fields

Abstract

We study global bifurcations in generic three-parameter families of vector fields on S 2 . In the recent article by Ilyashenko et al (2018 Invent. Math. 213 461-506), the authors show that three-parameter unfoldings of vector fields with the polycycle 'tears of the heart' are structurally unstable. We consider three-parameter unfoldings of vector fields with separatrix graphs 'ears' and 'glasses', and prove that these families are structurally unstable as well. We also study in more details the classical bifurcation of a saddle loop, and use it as a building block in our main example.

Families of vector fields with many numerical invariants

“…A vector field with a polycycle "tears of the heart". This figure was first published in Nonlinearity [14]. A similar figure was earlier published in Invent.…”

“…This figure was first published in Nonlinearity [14]. In [14], N. Solodovnikov and we prove that 3-parameter unfoldings of vector fields with "ears" and "glasses" separatrix graphs (see Fig. 2) are also structurally unstable, and the classification of these unfoldings has numerical invariants.…”

“…Vector fields with structurally unstable unfoldings from [14]. This figure was first published in Nonlinearity [14]. In [14], N. Solodovnikov and we prove that 3-parameter unfoldings of vector fields with "ears" and "glasses" separatrix graphs (see Fig.…”

“…Moderate equivalence respects the descriptions of bifurcations mentioned above. Moderate equivalence and other similar equivalences were used in [12,13,14,15,17]. For a comparison of different equivalence relations on the space of families of vector fields see [11].…”

“…[17]. Vector fields with structurally unstable unfoldings from [14]. This figure was first published in Nonlinearity [14].…”