Publication: A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields
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Issued Date
2016
Resource Type
File Type
application/pdf
ISSN
9388974
Other identifier(s)
2-s2.0-84976370597
Rights Holder(s)
Scopus
Bibliographic Citation
Journal of Nonlinear Science. Vol 26, No.6 (2016), p.1789-1815
Suggested Citation
Domokos G., Holmes P., Lángi Z. A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields. Journal of Nonlinear Science. Vol 26, No.6 (2016), p.1789-1815. doi:10.1007/s00332-016-9319-4 Retrieved from: https://hdl.handle.net/20.500.14740/4935
Author(s)
Abstract
Three-dimensional convex bodies can be classified in terms of the number and stability types of critical points on which they can balance at rest on a horizontal plane. For typical bodies, these are non-degenerate maxima, minima, and saddle points, the numbers of which provide a primary classification. Secondary and tertiary classifications use graphs to describe orbits connecting these critical points in the gradient vector field associated with each body. In previous work, it was shown that these classifications are complete in that no class is empty. Here, we construct 1- and 2-parameter families of convex bodies connecting members of adjacent primary and secondary classes and show that transitions between them can be realized by codimension 1 saddle-node and saddle–saddle (heteroclinic) bifurcations in the gradient vector fields. Our results indicate that all combinatorially possible transitions can be realized in physical shape evolution processes, e.g., by abrasion of sedimentary particles. © 2016, Springer Science+Business Media New York.