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

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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.

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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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Authors and Affiliations

  1. Department of Mechanics, Materials and Structures, Budapest University of Technology, Műegyetem rakpart 1-3., Budapest, 1111, Hungary

    Gábor Domokos

  2. Program in Applied and Computational Mathematics and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 08544, USA

    Philip Holmes

  3. Department of Geometry, Budapest University of Technology, Egry József u. 1., Budapest, 1111, Hungary

    Zsolt Lángi

Authors
  1. Gábor Domokos
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  2. Philip Holmes
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  3. Zsolt Lángi
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Correspondence to Zsolt Lángi.

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Communicated by Alain Goriely.

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Domokos, G., Holmes, P. & Lángi, Z. A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields. J Nonlinear Sci 26, 1789–1815 (2016). https://doi.org/10.1007/s00332-016-9319-4

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  • DOI: https://doi.org/10.1007/s00332-016-9319-4

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