2004/05/31 by Jean-Pierre Aubin, Jean‐Pierre Aubin, Annick Lesne
Environmental Science · Mathematics · Physics and Astronomy · Social Sciences · #Calculus (dental) #Computer science #Constructive #Differential (mechanical device) #Ecosystem dynamics and resilience #Energy landscape #Evolutionary Game Theory and Cooperation #Mathematics #Physics #Process (computing) #Set (abstract data type) #cond-mat.stat-mech
paper · pdf · doi:10.1063/1.1874332
28 pages, submitted to J. Math. Phys --
arxiv created 2004/09/08 · openalex publication_date 2005/03/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features “wells of duration T and altitude y.” We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells, and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) that have been developed during the last two decades; we therefore propose a minimal Appendix exposing them at the end of this paper to bridge the possible gap.