Cyclic characteristic

Oscillations describes this cyclic characteristic. There are three types of oscillations that can occur in a control loop. They are decreasing amplitude, constant amplitude, and increasing amplitude. Each is shown in Figure 10. 

Bykov et al., 1998 Lazman and Yablonskii, 1991 Lazman et al., 1985a) that the constant term Bq in Equation (25) is the non-zero multiple of the cyclic characteristic (33). 

The Horiuti numbers defining the cyclic characteristic in Equation (34) are relatively prime i.e. GCD(vi,..., v ) — 1. The exponent p in Equation (34) is the natural number. If we assume additionally that... 

Cyclic characteristic has following property (see Bykov et al., 1998, Corollary 14.2). 

We have found recently the topological interpretation of property (34). The stoichiometric constraints (24) can be interpreted in terms of the topological object, the circuit. Existence of the circuit "explains" the appearance of the cyclic characteristic in the constant term of kinetic polynomial. Thus, we can say that in some sense the correspondence between the detailed mechanism and thermodynamics is governed by pure topology. 

The cyclic characteristic C is small in the vicinity of thermodynamic equilibrium. We can find the overall reaction rate approximation in the vicinity of equilibrium either directly from kinetic polynomial or by expanding the reaction rate in power series by the small parameter C. The explicit expression for the first term is presented by Lazman and Yablonskii (1988, 1991). It is written as follows ... 

This linear affinity approximation does not always correspond to the linear approximation of kinetic polynomial - (Bo)/(Bj). This happens only when degree p of cyclic characteristic in Proposition 1 (see Equation (34)) is one. If p>l, linear approximation of the kinetic polynomial does not correspond to... 

Validity of the thermodynamic branch. To show that series (67) actually represents the thermodynamic branch we have to prove that all terms of this series include the cyclic characteristic C in positive degree. 

Proposition 3. If p> 1 and property (35) is valid, then the cyclic characteristic C is contained in the coefficient Bi of kinetic polynomial with an exponent equal... 

The interesting feature of our representation is that many sub-terms of the "fourth", non-linear, term may contain the "potential term" (the cyclic characteristic C) as well. It means that even in the domain "far from equilibrium" the open system still may have a "memory" about the equilibrium. Particular forms of this general Equation (77), i.e. for the cases of step limiting and the vicinity of equilibrium, respectively, are presented. 

The sense of the cyclic characteristic is simple. It is a kinetic equation of our brutto-reaction as if it were a step and consists of elementary reactions obeying the law of mass action. For the cycle with the brutto-equation 0 = 0, the cyclic characteristic is C = 0. If all cycles have the same "natural brutto-equations, their cyclic characteristics are represented as... 

Thus cyclic characteristics of various cycles will differ only in values of the factors (n ). Cyclic characteristics for two different cycles with the same"natural brutto-equations are proportional to each other... 

Mechanism I accounts for the "natural brutto-equation A = B obtained by adding steps of the detailed mechanism, whereas mechanisms II and III correspond to the equation 2 A = 2 B. Cyclic characteristics will, apparently, differ. In the former case C = K+ CA - K CB, in the latter C = K+Cl -K-Cl... 

As has been shown above, the cyclic characteristics is a kinetic equation for the brutto-reaction as if it were a simple step. But the denominator SjD, accounts for the "non-elementary character of this reaction and indicates the rate retardation by catalyst surface intermediates. 

In the former case, a step of each cycle can enter into one simple cycle. In the numerator of eqn. (46) for the step rate we will observe only one cyclic characteristic, C, corresponding to this cycle. The presence of an additional cycle affects only the value of the matching parameter P. The cycle rate can vary only quantitatively, but in neither case does the reaction direction vary. This situation corresponds to the so-called "kinetic matching (see, for example, ref. 40). Assuming that all steps are reversible, the total number of spanning trees amounts to n n2 + n,n —nxn2, where n, and n2 are the number of steps in both cycles. 

As an example, let us consider the above fragment of the conversion mechanism for n-hexane [its graph is given in Fig. 3(f)]. The weights of some arcs are equal to the sums of those of individual reactions. For example, the weight of the arc from HK to K amounts to bHK K = b1 + b5 + b6. Let us write down the rate for step (3). It enters into four cycles (see Fig. 4). Cycle I (HK MCK IK HK) has the cyclic characteristics... 

The cyclic characteristics and matching parameters for the cycles being known, we can easily determine a numerator for the steady-state rate. Since the denominator is cumbersome, we omit its description here. 

Let two cycles have similar "natural brutto-equations. Then their cyclic characteristics will be expressed as... 

Let us note one special but widespread case when there are several cycles but they have only common nodes (intermediates). Each cycle has its "own observed substance that is consumed or formed only in this cycle. The rate of concentration variation for this substance will have only one cyclic characteristic in the numerator, hence its expression by eqn. (59) is valid. 

The brutto-equation depends on the structure of the kinetic equation and its parameters. In Sect. 2.3 we have already spoken about cyclic characteristics in the numerator of the steady-state kinetic eqn. (46). It is the kinetic equation of the brutto-reaction as if it were a simple step. The form of the cyclic characteristics is independent of the detailed mechanism. But under... 

The numerator in equation (39) is referred to as the cyclic characteristics It represents the kinetic equation for the net reac tion with all its steps obeying the Law of Mass Action, whereas the denominator reflects the rate decrease caused by the reagents and the products. 

The cyclic characteristics are represented by the difference between the products of the weights of the steps participating... 

The numerator of the kinetic equation will consist of a cyclic characteristic and a conjugation parameter. 

Let us determine the rate of step 4 for the above mechanism. This rate is denoted in the graph by arc 32. From the diagonal element and of matrix H (92), we see that arc 32 appears in cycles 323 and 3213. The cyclic characteristics will be respectively ... 

The step participates in several cycles, and all cycles but one are stationary and their cyclic characteristics are equal to zero. The step participates in simple cycles having the same net equa tions and proportional cyclic characteristics. 

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