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Simplest bifurcation diagrams for monotone families of vector fields on a torus
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Baesens, Claude and MacKay, Robert S. (2018) Simplest bifurcation diagrams for monotone families of vector fields on a torus. Nonlinearity, 31 (6). 2928. doi:10.1088/1361-6544/aab6e2 ISSN 0951-7715.
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Official URL: https://doi.org/10.1088/1361-6544/aab6e2
Abstract
In Part 1 we prove that the bifurcation diagram for a monotone twoparameter family of vector fields on a torus has to be at least as complicated as the conjectured simplest one proposed in [BGKM1]. To achieve this we define “simplest” by minimising sequentially the numbers of equilibria, Bogdanov-Takens points, closed curves of centre and of neutral saddle, intersections of curves of centre and neutral saddle, Reeb components, other invariant annuli, arcs of rotational homoclinic bifurcation of horizontal homotopy type, necklace points, contractible periodic orbits, points of neutral horizontal homoclinic bifurcation and half-plane fan points. We obtain two types of simplest case, including that initially proposed. In Part 2 we analyse the bifurcation diagram for an explicit monotone family of vector fields on a torus and prove that it has at most two equilibria, precisely four Bogdanov-Takens points, no closed curves of centre nor closed curves of neutral saddle, at most two Reeb components, precisely four arcs of rotational homoclinic connection of “horizontal” homotopy type, eight horizontal saddle-node loop points, two necklace points, four points of neutral horizontal homoclinic connection, and two half-plane fan points, and there is no simultaneous existence of centre and neutral saddle, nor contractible homoclinic connection to a neutral saddle. Furthermore we prove that all saddle-nodes, Bogdanov-Takens points, non-neutral and neutral horizontal homoclinic bifurcations are non-degenerate and the Hopf condition is satisfied for all centres. We also find it has four points of degenerate Hopf bifurcation. Thus it provides an example of a family satisfying all the assumptions of Part 1 except the one of at most one contractible periodic orbit.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Library of Congress Subject Headings (LCSH): | Vector fields, Bifurcation theory, Torus (Geometry) | ||||||
Journal or Publication Title: | Nonlinearity | ||||||
Publisher: | Institute of Physics Publishing Ltd. | ||||||
ISSN: | 0951-7715 | ||||||
Official Date: | 8 May 2018 | ||||||
Dates: |
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Volume: | 31 | ||||||
Number: | 6 | ||||||
Article Number: | 2928 | ||||||
DOI: | 10.1088/1361-6544/aab6e2 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 1 March 2018 | ||||||
Date of first compliant Open Access: | 28 September 2018 |
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