Eigenvectors auxiliary system

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 general, coordinates of eigenvectors and r are simultaneously nonzero only for one value j — i because the auxiliary system is acyclic. However, Z d = 0 if just because that are eigenvectors for different eigenvalues, k, and kj. Hence, / d = 5ij. 

For left eigenvector V coordinate could have nonzero value only if there exists such q O that — i (this q is unique because the auxiliary dynamical system has no cycles). In that case (for q > 0),... 

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). 

For kinetic systems with well-separated constants the left and right eigenvectors can be explicitly estimated. Their coordinates are close to +1 or 0. We analyzed these estimates first for linear chains and cycles (5) and then for general acyclic auxiliary dynamical systems (34), (36) (35), (37). The distribution of zeros and +1 in the eigenvectors components depends on the rate constant ordering and may be rather surprising. Perhaps, the simplest example gives the asymptotic equivalence (for of the reaction network A,+2 with... 

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