# A stratum approach to global stability of complex balanced systems

@article{Siegel2010ASA, title={A stratum approach to global stability of complex balanced systems}, author={David Siegel and Matthew D. Johnston}, journal={Dynamical Systems}, year={2010}, volume={26}, pages={125 - 146} }

It has long been known that complex balanced mass-action systems exhibit a restrictive form of behaviour known as locally stable dynamics. This means that within each compatibility class – the forward invariant space where solutions lies – there is exactly one equilibrium concentration and that this concentration is locally asymptotically stable. It has also been conjectured that this stability extends globally to . That is to say, all solutions originating in approach the unique positive…

## 14 Citations

### Complex-balanced equilibria of generalized mass-action systems: necessary conditions for linear stability.

- MathematicsMathematical biosciences and engineering : MBE
- 2019

It is shown that linear stability (on the stoichiometric subspace and for all rate constants) implies uniqueness, and that, for classical mass-action systems, complex-balanced equilibria are not just asymptotically stable, but even diagonally stable (and hence linearly stable).

### Introduction to the Global Attractor Conjecture

- Mathematics
- 2016

The roots of the Global Attractor Conjecture (GAC) can be found in the 1972 paper “General Mass Action Kinetics” by Fritz Horn and Roy Jackson [9]. The paper is noted for presenting a general…

### A Proof of the Global Attractor Conjecture in the Single Linkage Class Case

- MathematicsSIAM J. Appl. Math.
- 2011

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.

### Weak Dynamical Nonemptiability and Persistence of Chemical Kinetics Systems

- MathematicsSIAM J. Appl. Math.
- 2011

This paper focuses on a result which states that, for a conservative mass-action system, persistence holds if every critical semilocking set is dynamically nonemptiable and the system contains no nested locking sets.

### A Geometric Approach to the Global Attractor Conjecture

- MathematicsSIAM J. Appl. Dyn. Syst.
- 2014

The main result states that the global attractor conjecture holds for complex-balanced systems that are strongly endotactic: every trajectory with positive initial condition converges to the unique positive equilibrium allowed by conservation laws.

### Topics in Chemical Reaction Network Theory

- Mathematics
- 2011

This thesis focuses on three topics within the scope of Chemical Reaction Network theory, and calls the theory Linear Conjugacy theory, which states that under the mass-action assumption two reaction networks with disparate topological structure may give rise to the same set of differential equations and therefore exhibit the same qualitative dynamical behaviour.

### On the Persistence and Global Stability of Mass-Action Systems

- MathematicsSIAM J. Math. Anal.
- 2012

The persistence of a large class of networks called lower-endotactic is studied and it is shown that in weakly reversible mass-action systems with two-dimensional stoichiometric subspace all bounded trajectories are persistent.

### Dynamical Equivalence and Linear Conjugacy of Chemical Reaction Networks: New Results and Methods

- Mathematics
- 2011

In the first part of this paper, we propose new optimization-based methods for the computation of preferred (dense, sparse, reversible, detailed and complex balanced) linearly conjugate reaction…

### Boundedness of trajectories for weakly reversible, single linkage class reaction systems

- Mathematics
- 2011

It is conjecture that the result holds when the reaction network is weakly reversible, and it is proved that this conjecture holds in the case when the Reaction network consists of a single linkage class, or connected component.

### Reaction Kinetics Path Based on Entropy Production Rate and Its Relevance to Low-Dimensional Manifolds

- PhysicsEntropy
- 2014

The equation is for the path of steepest descent that sufficiently accounts for the constraints inherent in chemical Kinetics such as element conservation, whereas the simple steepest-descent-path formulation cannot even approximately reproduce any part of the 1-D manifold of true chemical kinetics except for the special case where the eigenvector of the Hessian is nearly identical to that of the Jacobian of chemical kinetic.

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