A genealogy of convex solids via local and global bifurcations of gradient vector fields

摘要: Three-dimensional convex bodies can be classified in terms of the number and stability types critical points on which they balance at rest a horizontal plane. For typical these are nondegenerate maxima, minima, saddle-points, numbers provide primary classification. Secondary tertiary classifications use graphs to describe orbits connecting gradient vector field associated with each body. In previous work it was shown that complete no class is empty. Here we construct 1- 2-parameter families members adjacent secondary classes show transitions between them realized by codimension 1 saddle-node saddle-saddle (heteroclinic) bifurcations fields. Our results indicate all combinatorially possible physical shape evolution processes, e.g. abrasion sedimentary particles.