Criterion for uniqueness and multiplicity associated with the mechanism structure

4 CRITERION FOR UNIQUENESS AND MULTIPLICITY ASSOCIATED WITH THE MECHANISM STRUCTURE 

There is no doubt that studies for the establishment of new classes of mechanisms possessing an unique and stable steady state are essential and promising. On the other hand, it is of interest to construct a criterion for uniqueness and multiplicity that would permit us to analyze any reaction mechanism. An important contribution here has been made by Ivanova [5]. Using the Clark approach [59], she has formulated sufficiently general conditions for the uniqueness of steady states in a balance polyhedron in terms of the graph theory. In accordance with ref. 5 we will present a brief summary of these results. As before, we proceed from the validity of the law of mass action and its analog, the law of acting surfaces. Let us also assume that a linear law of conservation is unique (the law of conservation of the amount of catalyst). 

In accordance with ref. 5, a complex reaction is described by a so-called bipartite graph of its mechanisms having vertices (points) of two types. Type Z corresponds to substances Z, and type R is ascribed to the reaction Rp(i = 1, 2,. . ., n p = 1,. . ., N). Edges (Rp, Z ) and (Z , Rp) have weights 

In Sect. 1.3 we described cycles of two types, oriented and non-oriented. The oriented cycle can be passed by if we move in the direction of the arrows. For a cycle of a general type it can be different since it is a sequence of the vertices Z1(.. ., Zk where the pairs of vertices Zt and Zi+1(i = 1,.. ., /e-l)and also of Zk and Z, are connected by edges. As usual, we will consider simple cycles with no edge and no Z vertex appearing twice. 

To study the problem concerning the uniqueness and multiplicity of steady states it is necessary to consider one more type of cycle that is more general compared with oriented cycles. We will call them Clark (or Clark-Ivanova) cycles. 

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