A Geometric Approach to the Global Attractor Conjecture Academic Article uri icon

abstract

  • 2014 Society for Industrial and Applied Mathematics. This paper introduces the class of strongly endotactic networks, a subclass of the endotactic networks introduced by Craciun, Nazarov, and Pantea. The main result states that the global attractor conjecture holds for complex-balanced systems that are strongly endotactic: every trajectory with positive initial condition converges to the unique positive equilibrium allowed by conservation laws. This extends a recent result by Anderson for systems where the reaction diagram has only one linkage class (connected component). The results here are proved using differential inclusions, a setting that includes power-law systems. The key ideas include a perspective on reaction kinetics in terms of combinatorial geometry of reaction diagrams, a projection argument that enables analysis of a given system in terms of systems with lower dimension, and an extension of Birch's theorem, a well-known result about intersections of affine subspaces with manifolds parameterized by monomials.

published proceedings

  • SIAM Journal on Applied Dynamical Systems

altmetric score

  • 4

author list (cited authors)

  • Gopalkrishnan, M., Miller, E., & Shiu, A.

citation count

  • 51

complete list of authors

  • Gopalkrishnan, Manoj||Miller, Ezra||Shiu, Anne

publication date

  • January 2014