This paper is concerned with the dynamical properties of deterministically modeled chemical reaction systems. Specifically, this paper provides a proof of the Global Attractor Conjecture in the setting where the underlying reaction diagram consists of a single linkage class, or connected component. The conjecture dates back to the early 1970s and is the most well known and important open problem in the field of chemical reaction network theory. The resolution of the conjecture has important biological and mathematical implications in both the deterministic and stochastic settings. The method of partitioning a set of vectors along a sequence is introduced and acts as one of the main analytical tools throughout .
This paper is concerned with the dynamical properties of deterministically modeled chemical reaction systems. Specifically, this paper provides a proof of the Global Attractor Conjecture in the setting where the underlying reaction diagram consists of a single linkage class, or connected component. The conjecture dates back to the early 1970s and is the most well known and important open problem in the field of chemical reaction network theory. The resolution of the conjecture has important biological and mathematical implications in both the deterministic and stochastic settings. One of our main analytical tools, which is introduced here, will be a method for partitioning the relevant monomials of the dynamical system along sequences of trajectory points into classes with comparable growths. This will allow us to conclude that if a trajectory converges to the boundary, then a whole family of Lyapunov functions decrease along the trajectory, a fact that will allow us to overcome the fact that the usual Lyapunov function used in this setting does not diverge to infinity as trajectories converge to the boundary, which has been the technical sticking point to a proof of the Global Attractor Conjecture over the years .
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The method of partitioning a set of vectors along a sequence is introduced and acts as one of the main analytical tools throughout
One of our main analytical tools, which is introduced here, will be a method for partitioning the relevant monomials of the dynamical system along sequences of trajectory points into classes with comparable growths. This will allow us to conclude that if a trajectory converges to the boundary, then a whole family of Lyapunov functions decrease along the trajectory, a fact that will allow us to overcome the fact that the usual Lyapunov function used in this setting does not diverge to infinity as trajectories converge to the boundary, which has been the technical sticking point to a proof of the Global Attractor Conjecture over the years