Zero-deficiency theorem

The zero deficiency theorem Lett us consider the reaction... 

By now all the notions are at hand to state the zero deficiency theorem by Feinberg, Horn and Jackson. 

It is obvious that a concentration x for which (4.13) holds is an equilibrium concentration. It is not so obvious that mechanisms with the above property are of deficiency zero. This fact is an important by-product of the Feinberg-Horn-Jackson theory and shows that the zero deficiency theorem expresses a generalisation of classical beliefs. 

Statement (3) together with the zero deficiency theorem implies all the other statements, as the deficiency of detailed balanced reactions is zero. 

As the V-graph of a weakly reversible mechanism is cyclic (Exercise 4 below), the regular behaviour of a mechanism cannot be inferred from both zero deficiency theorem and by Vol pert s theorem at the same time at most one of them can be used. 

The examples given above were mostly constructed upon the basis of deep qualitative knowledge of the behaviour of the trajectories of the individual differential equations. Some papers suggest that, following their methods, the reader is able to construct an unlimited number of chaotic kinetic models. Still, there is a desire to obtain information more easily. This means that statements like the zero deficiency theorem are needed that assure or exclude chaotic behaviour using only knowledge on the algebraic structure of the complex chemical reaction. So far only small steps have been taken in this direction. 

One of the first to recognize the power of network approaches was Feinberg, a chemical engineer. As well as an important result, he and his collaborators derived a useful formalism, known as the zero-deficiency theorem, which we summarize... 

The package does not do Hopf bifurcation analysis nor have any direct way to distinguishing between limit cycle and chaotic attractors. The package contains the Zero Deficiency Theorem, the "knot tree network theorem" as well as some older theorems that identify stable networks. The package solves the general reaction balancing problem whose solution is a convex polyhedral cone of extreme reactions. It handles thermodynamic properties of reactions assuming ideality. 

The first part of the theorem means that within the class of zero deficiency mechanisms weak reversibility is enough to ensure regular behaviour. 

Next, the deficiency of the augmented network is calculated (with 0 a complex which, by definition, is compatible with A). If the deficiency is zero, the strong deficiency zero theorem (Feinberg, 1987) applies Provided the kinetics are of the mass action type, no matter what (positive) values the kinetic constants may have, the CSTR cannot exhibit multiple steady states, unstable steady states, or periodic orbits. The result is, in a sense, very strong because the governing differential... 

Feinberg, M., Chemical reaction network structure and the stability of complex isothermal reactors. I, The deficiency zero and deficiency one theorems. Chem. Eng. Sci. 42,2229 (1987). 

Feinberg, M. Complex balancing in general kinetic systems. Arch. Rat. Mech. Anal. 1972, 49, 187-194 Chemical-reaction network structure and the stability of complex isothermal reactors. 1. The deficiency-zero and deficiency-one theorems. Chem. Eng. Set 1987, 42, 2229-2268 Chemical-reaction network structure and the stability of complex isothermal reactors. 2. Multiple steady-states for networks of deficiency one. Chem. Eng. Set 1988, 43, 1-25. 

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