On the structural characterisation of chaotic chemical reactions

The examples given above were mostly constructed upon the basis of deep qualitative knowledge of the behaviour of the trajectories of the individual differential equations. Some papers suggest that, following their methods, the reader is able to construct an unlimited number of chaotic kinetic models. Still, there is a desire to obtain information more easily. This means that statements like the zero deficiency theorem are needed that assure or exclude chaotic behaviour using only knowledge on the algebraic structure of the complex chemical reaction. So far only small steps have been taken in this direction. 

Another kind of approach has been initiated in the recent investigations by Toth and Hars (1986a). They studied linear transforms of the Lorenz equation (1968) and of the Rossler model (1976) in order to obtain kinetic models. The failure of their efforts underline the importance of negative cross-effects. Based upon these results the following conjecture can be formulated if a nonkinetic polynomial differential equation shows chaotic behaviour then it cannot be transformed into a kinetic one. 

Show that none of the proper and improper orthogonal transforms of the Lorenz equation are kinetic equations. 

Hint to solution 1. The general form of the proper orthogonal transformations can be found in Section 14.10 of Korn Korn (1968). The proof based on this is contained in Toth Hars (1986a). 

Hint to solution 2. Apply the theory of algebraic invariants of differential equations as outlined, for example, in the book by Sibirskii (1982, p. 91 et seq.). 

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