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

G. 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)1-27
Number of pages27
JournalJournal of Nonlinear Science
DOIs
Publication statusAccepted/In press - Jun 29 2016

Fingerprint

Genealogy
Gradient vector
Local Bifurcations
Global Bifurcation
Vector Field
Convex Body
Critical point
Connecting Orbits
Saddle
Saddlepoint
Abrasion
Codimension
Orbits
Horizontal
Bifurcation
Adjacent
Three-dimensional
Graph in graph theory
Vertex of a graph
Class

Keywords

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

ASJC Scopus subject areas

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

Cite this

A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields. / Domokos, G.; Holmes, Philip; Lángi, Zsolt.

In: Journal of Nonlinear Science, 29.06.2016, p. 1-27.

Research output: Contribution to journalArticle

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