## CNp qa21d3 2d1 N d32 N3 qa21d3 a2di 202

These equilibriums correspond to the chains with lengths 1, 2, and 3, respectively. The first chain is stable if q < d1d2/a1, the second chain is stable if d1d2/a1 < q < d1d2/a1 + d2d3/a2, and the third is stable if q > d1d2/a1 + d2d3/a2. By the same token the axis q is divided into non-overlapping intervals, in each of them a chain of fixed length can exist (see Fig. 7.4). The general result for any length was described in Svirezhev and Logofet (1978).

If we assume that the main statement of Prigogine's theorem about the minimisation of the internal entropy production in the stable equilibrium takes place in our case, then we have to prove that the value s = d1N1 + d2N2 + d3N3 is minimal in each stable equilibrium of Eq. (5.2). Fig. 7.3. To the proof of the s minimality. N2 = f(N1) = (1 /a1 )(q/N1 - d1), Nip (a) = q/d1, Nip (b) = d2/a1, N(b) = (q/d2) - (d1 /a1); (a) q < qj, (b) q > qj.

Let q < q1 = d1 d2 /a1, then only the state (a) that lies at the point of intersection of isoclines N2 = (1/a1 )(q/N1 - d1) and N2 = 0 (Fig. 7.3a) is stable.

If we could prove that moving along the first isocline where dN1 /dt = 0 to the equilibrium Nj (a) = q/d1, we decrease the value of s, then, by the same token, we would prove that sis minimal at this point. The proof is elementary. Indeed, along the first isocline d2 1 d1 d2 s = d1 N 1 + q w - , and, if N < Nj(a), q < d1 d2/a1, then 8s/8NV1 < 0.

Now let d1 d2/a1 + d2d3 /a2 = qj > q > q1 = d1 d2/a1, then only the state (b), which lies at the point of intersection of isoclines N2 = (1/a1 )(q/N1 — d1), N1 = d2/a1 and N3 = 0 (Fig. 7.3b) is stable. In this case, although we shall move along the first isocline in the opposite direction from point Nj (a) = q/d1 (which now becomes unstable) towards the stable equilibrium (b), the value of s will again decrease, reaching its minimum at equilibrium. Analogously we can prove the same statement for the equilibrium (c).

We have thus proved some analogue of Prigogine's theorem about the minimisation of the internal entropy production in the stable equilibrium (this statement can be proved for any n). However, in spite of the fact that the formulations of Prigogine's original theorem and our Prigogine-like theorem are very similar, they differ significantly from one another. If the original theorem was proved only for the system with a single dynamic equilibrium, then our Prigogine-like theorem deals with a system possessing several multiple equilibriums, which replace each other as a result of pre-determined bifurcations. Let us consider this process in more detail.

Assume that the value of inflow q slowly increases, so that one of these equilibriums is established rather quickly in relation to the "slow" time. If we now calculate the dynamics of the internal entropy production in relation to "slow" time, i.e. in the equilibrium, then