Math @ Duke

Publications [#243889] of Ezra Miller
Papers Accepted
 with Gopalkrishnan, M; Miller, E; Shiu, A, A Geometric Approach to the Global Attractor Conjecture,
SIAM Journal on Applied Dynamical Systems, vol. 13 no. 2
(2013),
pp. 758797 (49 pages.) [math.DS/1305.5303], [doi]
(last updated on 2018/05/21)
Abstract: This paper introduces the class of "strongly endotactic networks", a subclass
of the endotactic networks introduced by G. Craciun, F. Nazarov, and C. Pantea.
The main result states that the global attractor conjecture holds for
complexbalanced 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 D. F. 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 powerlaw 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 wellknown result
about intersections of affine subspaces with manifolds parameterized by
monomials.


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