Catastrophe transcritical bifurcation

The saddle node catastrophe and the Hopf bifurcation may be shown to be structurally stable. Certain additional conditions (see Sections 5.5.2.2, 5.5.2.3) are imposed on the transcritical bifurcation and the pitchfork bifurcation. The system is structurally stable under perturbations not disturbing these additional conditions on the other hand, when arbitrary... 

Note that equation (6.28) has a form of the transcritical bifurcation, (5.69). Equation (5.69) may describe the catastrophe occurring on a change of sign of the parameter a. However, an analogous coefficient in equation (6.28), k, is always non-negative. Therefore, in the chemical system described by (6.28) a catastrophe cannot take place. The stationary state Xj = 0 is unstable and the state x2 = kla/k 1 is stable in the entire range of variability of the control parameters klt k t, a. 

Equation (6.30) has a standard form of the transcritical bifurcation, see (5.69). A transcritical catastrophe may occur since the coefficient k2a — k2 may change sign. However, properties of the system (6.30) differ somewhat from those of a general system (5.69). Note that the stationary states of kinetic equation (6.30) are given by... 

When kta — k2 < 0, the system has just one stationary state, xx = 0, which is stable. In contrast, when k1a — k2> 0, the system has two stationary states Xj = 0, an unstable state, and x2 = (k1a — k2)/k l, a stable state. Recall that in a standard system of the transcritical bifurcation (5.69) the catastrophe of stability exchange between the two existing states Xj and x2 takes place. On the other hand, in the chemical system (6.30) the catastrophe involves destabilization of the state x1 = 0 with a simultaneous appearance of the stable state x2 > 0. 

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See also in sourсe #XX -- [ , ]

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