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Mathematics > Dynamical Systems

arXiv:1804.04596 (math)
[Submitted on 12 Apr 2018 (v1), last revised 8 Jan 2019 (this version, v3)]

Title:Large bifurcation supports

Authors:Nataliya Goncharuk, Yulij Ilyashenko
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Abstract:In the study of global bifurcations of vector fields on $S^2$, it is important to distinguish a set "where the bifurcation actually occurs", -- the bifurcation support. Hopefully, it is sufficient to study the bifurcation in a neighborhood of the support only.
The first definition of bifurcation support was proposed by this http URL. However this set appears to be too small. In particular, the newly discovered effect, an open domain in the space of three-parametric families on $S^2$ with no structurally stable families, is not visible in a neighborhood of the bifurcation support.
In this article, we give a new definition of "large bifurcation support" that accomplishes the task. Roughly speaking, if we know the topological type of the phase portrait of a vector field, and we also know the bifurcation in a neighborhood of the large bifurcation support, then we know the bifurcation on the whole sphere.
Subjects: Dynamical Systems (math.DS)
MSC classes: 34C23, 37G99, 37E35
Cite as: arXiv:1804.04596 [math.DS]
  (or arXiv:1804.04596v3 [math.DS] for this version)
  https://doi.org/10.48550/arXiv.1804.04596

Submission history

From: Nataliya Goncharuk Mrs [view email]
[v1] Thu, 12 Apr 2018 16:06:10 UTC (429 KB)
[v2] Thu, 13 Sep 2018 15:28:33 UTC (449 KB)
[v3] Tue, 8 Jan 2019 01:17:36 UTC (450 KB)
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