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Germs of bifurcation diagrams and SN-SN families. (English) Zbl 1466.37042

The author studies the geometry of bifurcation diagrams of families of vector fields in the plane. A family is “glocal” if it is local in the parameters and global in the phase variable. The author presents the following counterexample to the Arnold’s conjecture about bifurcation diagrams: there exists a countable number of two-parameter glocal families whose germs of simple bifurcation diagrams are pairwise topologically non-equivalent. These families (which are structurally stable) are of class SN-SN: they are unfoldings of codimension two degeneracies of vector fields that have one saddle-node singular point and one saddle-node limit cycle, both of multiplicity. Previously, this effect was discovered for three parameters only. One of the tools used in the proof of structural stability is an embedding theorem for saddle-node families depending on a parameter.

MSC:

37G05 Normal forms for dynamical systems
37C20 Generic properties, structural stability of dynamical systems
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
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