Toric Differential Inclusions and a Proof of the Global Attractor Conjecture

By Gheorghe Craciun. 2015

114 citations recorded by the source corpus.

The global attractor conjecture says that toric dynamical systems (i.e., a class of polynomial dynamical systems on the positive orthant) have a globally attracting point within each positive linear invariant subspace -- or, equivalently, complex balanced mass-action systems have a globally attracting point within each positive stoichiometric compatibility class. A proof of this conjecture implies that a large class of nonlinear dynamical systems on the positive orthant have very simple and stable dynamics. The conjecture originates from the 1972 breakthrough work by Fritz Horn and Roy Jackson, and was formulated in its current form by Horn in 1974. We introduce toric differential inclusions, and we show that each positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the origin. We use this result to prove the global attractor conjecture. In particular, it follows that all detailed balanced mass action systems and all deficiency zero weakly reversible networks have the global attractor property.

Subfields: Geometry and Topology, Molecular Biology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/dcd594c83157a93b239a32b90d589fc8df1bd916

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Polly Y. Yu
2. Riccardo Rao
3. Massimiliano Esposito

In Molecular Biology

1. Polly Y. Yu
2. Riccardo Rao
3. Massimiliano Esposito
4. Gheorghe Craciun
5. Paul C. Bressloff

In Statistical and Nonlinear Physics

1. Riccardo Rao
2. Polly Y. Yu
3. Massimiliano Esposito

Citations in the corpus

• Nonequilibrium Thermodynamics of Chemical Reaction Networks: Wisdom from Stochastic Thermodynamics (2016, background): “The complex-balanced distribution (37) depends on the unbroken components {Lu}, which can be inferred from the initial conditions, and is always globally stable [68].”
• Mathematical Analysis of Chemical Reaction Systems (2018, background): “A proof of the global attractor conjecture in full generality has been recently proposed in [17].”