Attractors auxiliary discrete dynamical system

If the reader can use these properties (when it is necessary) without additional clarification, it is possible to skip reading Section 3 and go directly to more applied sections. In Section 4 we study static and dynamic properties of linear multiscale reaction networks. An important instrument for that study is a hierarchy of auxiliary discrete dynamical system. Let A, be nodes of the network ("components"), Ai Aj be edges (reactions), and fcy,- be the constants of these reactions (please pay attention to the inverse order of subscripts). A discrete dynamical system

dynamical system for a given network we find for each A,- the maximal constant of reactions Ai Af k ( i)i>kji for all j, and — i if there are no reactions Ai Aj. Attractors in this discrete dynamical system are cycles and fixed points. 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

In the simplest case, the auxiliary discrete dynamical system for the reaction network W is acyclic and has only one attractor, a fixed point. Let this point be A (n is the number of vertices). The correspondent eigenvectors for zero eigenvalue are r = S j and Z = 1. For such a system, it is easy to find explicit analytic solution of kinetic equation (32). 

The second simple particular case on the way to general case is a reaction network with components A, ..., A whose auxiliary discrete dynamical system has one attractor, a cycle with period t > 1 A +i A - +x. ., A ... 

Again we should analyze, whether this new cycle is a sink in the new reaction network, etc. Finally, after a chain of transformations, we should come to an auxiliary discrete dynamical system with one attractor, a cycle, that is the sink of the transformed whole reaction network. After that, we can find stationary distribution by restoring of glued cycles in auxiliary kinetic system and applying formulas (11)-(13) and (15) from Section 2. First, we find the stationary state of the cycle constructed on the last iteration, after that for each vertex Ay that is a glued cycle we know its concentration (the sum of all concentrations) and can find the stationary distribution, then if there remain some vertices that are glued cycles we find distribution of concentrations in these cycles, etc. At the end of this process we find all stationary concentrations with high accuracy, with probability close to one. 

For one catalytic cycle, relaxation in the subspace = 0 is approximated by relaxation of a chain that is produced from the cycle by cutting the limiting step (Section 2). For reaction networks under consideration (with one cyclic attractor in auxiliary discrete dynamical system) the direct generalization works for approximation of relaxation in the subspace = 0 it is sufficient to perform the following procedures ... 

The general case cycles surgery for auxiliary discrete dynamical system with arbitrary family of attractors... 

Figure 4 Four possible auxiliary dynamical systems for the reversible triangle of reactions with k2T>kij for (/,/)y (2,l) (a) kn>ki2, k23>k i (b) kn>ki2, kn>k2i (c) ki2>ku, k2i>kn and (d) ki2>k- 2, kii >/c23- Foi" each vertex the outgoing reaction with the largest rate constant is represented by the solid bold arrow, and other reactions are represented by the dashed arrows. The digraphs formed by solid bold arrows are the auxiliary discrete dynamical systems. Attractors of these systems are isolated in frames.

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