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

Gábor Domokos, Philip Holmes, Zsolt Lángi

Research output: Contribution to journalArticle

2 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)1789-1815
Number of pages27
JournalJournal of Nonlinear Science
Volume26
Issue number6
DOIs
Publication statusPublished - Dec 1 2016

Keywords

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

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering(all)
  • Applied Mathematics

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