The Number and Stability of Equilibrium States in Closed Systems

The Number and Stability of Equilibrium States in Closed Systems 

In Section 1.4 the topological concept of rotation was used to prove the existence of equilibrium states. When the reaction kinetics are not restricted by Postulate 1.5.1, each invariant manifold may include more than one equilibrium states and it is interesting to obtain information about the number and stability of these states. In the present section we shall use one more topological concept, the index of a fixed point, to show that the equilibrium states are odd in number, 2m+1, among which m at least are unstable. As in the preceding sections, the discussion concerns isolated systems, but extension to other closed systems should not present difficulties. 

An equilibrium state or point Gf(wo) is defined as a solution of the equation 

According to Theorems A.5, A.6, the calculation of the index of a point satisfying Eq. (1.7.2) is based on the eigenvalue problem 

Theorem 1.7.1. If the matrix A( ) is nonsingular for all equilibrium points Gf(Wo) then the number of these points is odd, n = 2m-l-l, among which m-h 1 have index (—1) and the remaining m have index 

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