Dissipative structures bifurcation theory

Kuhicek, M., and M. Marek. Computational Methods in Bifurcation Theory and Dissipative Structures, Springer-Verlag, Berhn (1983). 

The mathematical theory of dissipative structures is mainly based on approximate methods such as bifurcation theory of singular perturbation theory. Situations like those described in Section VI and that permit an exact solution are rather exceptional. 

Kubicek, M. Marek, M. 1983 In Computational methods in bifurcation theory and dissipative structures, pp. 39-47. New York Springer-Verlag. 

The steady-state solution that is an extension of the equilibrium state, called the thermodynamic branch, is stable until the parameter A reaches the critical value A,. For values larger than A<, there appear two new branches (61) and (62). Each of the new branches is stable, but the extrapolation of the thermodynamic branch (a ) is unstable. Using the mathematical methods of bifurcation theory, one can determine the point A, and also obtain the new solution, (i.e., the dissipative structures) in the vicinity of A, as a function of (A - A,.). One must emphasize that... 

At present, all the results of studies in this field cannot be interpreted if the knowledge of the systems under far from equilibrium conditions is neglected. Although stability studies and bifurcation theories, when applied to solid systems give new example of dissipative structures, these structures show certain specific characters. We shall illustrate this proposal by means of studies on oxidation of metals and alloys which produce a very interesting ordering of the corrosion scale. 

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