Uniqueness and Stability of Steady States in Open Systems

On the other hand, since —each point corresponds 

This relation contradicts the monotonically increasing character of the function T—6Q T)/VCp, which evaluated at T=0 gives Q(0) 0. The vector fields I —0f and I have now been shown to be homotopic and hence they have the same rotation, 4-1. This implies that Eqs. (1.8.27), (1.8.28) have one or more solutions in r Uf). 

Uniqueness and StabQity of Steady States in Open Systems 

We shall turn now to the stirred tank reactor, which has been already studied in Section 1.8 relative to the existence of steady states. In this section we shall show that the steady states are odd in number, 2 m 4-1, among which m at least are unstable. Moreover, for values of the parameter 6 outside a certain range, the steady state is unique and stable. This will be the extent of our occupation with the stability problem. For standard work on the dynamics and stability of stirred tank reactors, including detailed phase space analyses, see the textbook of Aris [3], the work of Aris and Amundson [6,7], and Luus and Lapidus [32]. 

To understand the exceptional case of 1/0 being an eigenvalue of the matrix A( J we should note that Eqs.(1.9.1) define implicitly a transformation from the variables 0,Cij.Cfff, Tf to the variables 

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