Butterfly catastrophe

Fig. 6 The butterfly catastrophe generates three states of the output variable x controlled by four control factors...

Actually there are two cusps which can interact in some cases (204) to form a higher order catastrophe called butterfly catastrophe because of the fancy shape of its bifurcation. 

The whole phenomenology of phase behavior and emulsion inversion was interpreted wifli a butterfly catastrophe model with amazing quahtative matching between theory and experiment. The phase behavior model used the Maxwell convention which allows the system to split into several states, i.e., phases at equilibrium. On the other hand, the emulsion-type model allows for only one state (emulsion type) at the time, with eventually catastrophic transition and hysteresis, according to the perfect delay convention. The fact that the same model potential permits the interpretation of the phase behavior and of the emulsion inver sion (204, 206) is a symptomatic hint that both phe-nomenologies are linked, probably through formulation and water/oil composition which are two of the four manipula-ble parameters in the butterfly catastrophe potential. 

The butterfly catastrophe model explains why the transitional inversion is not really an inversion but a surfactant transfer from one phase to the other, while the catastrophic inversion is a nonreversible hysteresis type instability. This approach, which is out of the scope of this chapter, is well documented elsewhere (197). 

JL Salager. Phase behavior of amphiphile-oil-water systems related to the butterfly catastrophe. J Colloid Interface Sci 105 21—26, 1985. 

By the way, both the phase behavior at equilibrium and the emulsion dynamic inversion features can be interpreted in a relatively simple way by a sixth-order catastrophe, the so-called butterfly model [56]. 

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