Primary bifurcation

Figure 5.21 Bifurcation far from equilibrium, (a) Primary bifurcation is the distance from equilibrium, at which the thermodynamic branching of minimal entropy production becomes unstable. The bifurcation point or critical point corresponds to the concentration (b) Complete diagram of bifurcations. As the non-linear reaction moves away from equilibrium, the number of possible states increases enormously. (Adapted, with permission, from Coveney and Highfield, 1990).

Primary bifurcation, the first transition from the reference state on the thermodynamic branch, was defined and discussed in the paper by I. Prigogine. This phenomenon is nowadays well understood. Let us outline briefly its theoretical formulation for the reaction-diffusion equations1-2... 

Fig. 3. Mirror-image polar patterns arising through a primary bifurcation.

Modifications of this scenario in the vicinity of the primary bifurcation point /X = 0 include the case where the cubic couplings are such that the stripes also arise subcritically. This is for instance the case [43] in a region of... 

Fig. 4. Bifurcation diagram for variable X of the Brusselator (A = 4.5, Dy/Dx = 8) as a function of the parameter B. It exhibits the standard hex-stripe competition (hysteresis loop) near the primary bifurcation (Be = 6.71). Reentrant hexagons become stable for higher values of the bifurcation parameter B. [Xmax — A] is represented.

This idea was proposed by Andronov and Leontovich in their first work [9] which deals with primary bifurcations of limit cycles on the plane. Further developments of the theory of bifurcations, internal to the Morse-Smale class, has also confirmed the sufficiency of using finite-parameter families for a rather large number of problems. 

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