# Locally stable dynamics

A chemical reaction network associated with some kinetic scheme is said to have **locally stable dynamics** if there exists within each positive stoichiometric compatibility class exactly one equilibrium concentration , and if this equilibrium concentration is locally asymptotically stable relative to its compatibility class. That is to say, trajectories within the invariant space as and *close enough* to it converge asymptotically to .

Many kinetic systems within chemical reaction network theory are known to exhibit locally stable dynamics, in particular, detailed^{[1]} and complex balanced^{[2]} mass action systems. It precludes such exotic behavior as multistability.

Locally stable dynamics can be naturally extended **globally stable dynamics** when the unique equilibrium concentration in each compatibility class is known to an attractor for the entire set . Extending locally stable dynamics to globally stable dynamics for complex balanced mass action systems is the content of the Global Attractor Conjecture, one of the longest-standing and best known conjectures in chemical reaction network theory ^{[3]}^{[4]}.

The term "locally stable dynamics" first appeared in the papers of David Siegel and Matthew D. Johnston in an attempt to standardize and simplify the terminology for what is a recurrent concept throughout chemical reaction network theory ^{[5]}^{[6]}. It is not otherwise widely used.

## References

- ↑ Aizik I. Vol'pert and Sergei I. Hudjaev,
*Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics*, Martinus Nijhoff Publishers, Dordrecht, Netherlands, 1985 - ↑ Fritz Horn and Roy Jackson, General mass action kinetics,
*Arch. Ration. Mech. Anal.*, 47:81--116, 1972 - ↑ Gheorghe Craciun, Alicia Dickenstein, Anne Shiu, and Bernd Sturmfels, Toric dynamical systems,
*J. Symbolic Comput.*, 44(11):1551--1565, 2009 - ↑ David Anderson, A proof of the global attractor conjecture in the single linkage class case,
*SIAM J. Appl. Math.*, 71(4):1487--1508, 2011 - ↑ Matthew D. Johnston and David Siegel, Linear conjugacy of chemical reaction networks,
*J. Math. Chem.*, 49(7):1263--1282, 2011 - ↑ David Siegel and Matthew D. Johnston, A stratum approach to global stability of complex balanced systems,
*Dyn. Syst.*, 26(2):125--146, 2011