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Moscow Mathematical Journal

Volume 20, Issue 2, April–June 2020  pp. 323–341.

Bifurcations of the Polycycle “Tears of the Heart”: Multiple Numerical Invariants

Authors:  Nataliya Goncharuk (1) and Yury Kudryashov (1)
Author institution:(1) Cornell University, Department of Mathematics, 310 Malott Hall, Ithaca, NY 14853-4201 USA


“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 a recent article by Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov, 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.

2010 Math. Subj. Class. 34C23, 37G99, 37E35.

Keywords: Planar vector fields, structural stability, polycycles, bifurcations.

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