MathDB

Global Attractor Conjecture

Let a mass-action reaction network with stoichiometric subspace S⊆RnS\subseteq\mathbb{R}^n be complex balanced, and let x˙=f(x)\dot{x}=f(x) be its mass-action system. For every initial condition x0∈R>0nx_0\in\mathbb{R}_{>0}^n, the stoichiometric compatibility class (x0+S)∩R>0n(x_0+S)\cap\mathbb{R}_{>0}^n contains a unique equilibrium x∗∈R>0nx^*\in\mathbb{R}_{>0}^n, and the corresponding trajectory satisfies lim⁡t→∞x(t)=x∗\lim_{t\to\infty}x(t)=x^*, where x(0)=x0x(0)=x_0.

References

Primary source

arXiv

Additional references

dynamical-systemsreaction-network-theory
Posted · edited

Progress summary

Refreshed
Claimed solved

A preprint claims a complete proof, but the newest result establishes convergence only under additional structural assumptions, so the general conjecture is not independently confirmed.

The conjecture concerns global convergence for complex-balanced reaction networks. It originated in work of Horn and Jackson in 1972 and was formulated in its current form by Horn in 1974.

Known results

  • Craciun, Nazarov, and Pantea proved the three-dimensional case.
  • Pantea handled invariant-subspace dimension at most 33.
  • Anderson proved the single-linkage-class case.
  • Gopalkrishnan, Miller, and Shiu proved the strongly endotactic case, extending earlier work on single-linkage-class systems.

September 2026 special-case result

Carsten Wiuf's A Proof of the Global Attractor Conjecture in a Special Case, reported on September 21, 2026, uses boundary-limit exclusion to establish global convergence under two structural hypotheses, including networks with multiple linkage classes. Separately, Toric Differential Inclusions and a Proof of the Global Attractor Conjecture claims a full proof, but the retrieved material provides no independent verification.

Current status (as of September 2026): A preprint claims the conjecture is proved in full, while the newest explicitly dated result covers only a structurally restricted class; the general claim remains unverified.

Sources

Solutions 0

No solutions have been posted yet.