We say that a polyhedral fan F covers R n if ⋃ C ∈ F C = R n. That cover R n, such that F is a polyhedral fan 13 13 13Ī polyhedral fan in R n is a finite set F of polyhedral cones such that ( i ) any face of a cone in F is also in F, and ( i i ) the intersection of two cones in F is a face of both cones. Consider a finite set F of polyhedral cones 12 12 12Ī polyhedral cone in R n is the intersection of a finite set of half-spaces of R n. Toric differential inclusions are defined as follows. īut soon afterwards Horn explained that they have not actually proved this claim, and in 1974 he proposed this global convergence property as a conjecture. I.e., if the positive equilibrium x 0 belongs to the linear invariant subspace S 0, then all trajectories that start in S 0 converge to x 0. Also in Horn and Jackson have shown that these systems enjoy remarkable stability properties, and in particular they have a unique positive equilibrium within each linear invariant subspace, and this equilibrium is locally asymptotically stable.Īctually, in Horn and Jackson stated that the unique positive equilibrium within each linear invariant subspace is a global attractor 9 9 9 The name “toric dynamical system” has been introduced recently to emphasize the remarkable algebraic properties of these systems, but this class of polynomial dynamical systems has been first studied in depth in the 1972 breakthrough paper of Horn and Jackson, where they have been called complex balanced mass-action systems. In particular, it follows that all detailed balanced mass action systems and all deficiency zero weakly reversible networks have the global attractor property. We use this result to prove the global attractor conjecture. 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. 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. 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.Ī proof of this conjecture implies that a large class of nonlinear dynamical systems on the positive orthant have very simple and stable dynamics.
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