Journal of Nonlinear Science

, Volume 26, Issue 6, pp 1789–1815 | Cite as

A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields

Article
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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.

Keywords

Codimension 2 bifurcation Convex body Equilibrium Morse–Smale complex Pebble shape Saddle-node bifurcation Saddle–saddle connection 

Mathematics Subject Classification

52A15 53A05 53Z05 
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Notes

Acknowledgments

This work was supported by OTKA Grant T119245 and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Comments from an anonymous referee and Tímea Szabó are gratefully acknowledged.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mechanics, Materials and StructuresBudapest University of TechnologyBudapestHungary
  2. 2.Program in Applied and Computational Mathematics and Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrincetonUSA
  3. 3.Department of GeometryBudapest University of TechnologyBudapestHungary

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