By Roger Chesneau
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Extra resources for Aeroguide 22 - General Dynamics F-111 E/ F
6 displays a cross-section of the bifurcation set of this fivedimensional structure for certain control variable values. It shows the "pocket of compromise," bounded by three distinct cusp points. As with the cusp catastrophe, Ct and C2 are normal and splitting factors respectively. Zeeman labels C3 the bias factor which tilts the initial cusp surface one way or another. C4 is the butterfly factor (not to be confused with the "butterfly effect" of chaos theory) that generates the pocket of compromise zone for certain of its values.
This does not disturb Thorn who sees qualitative understanding as at least as philosophically valuable as quantitative model building. Although the long-term trend has been to favor "neo-positive" quantitative model building, this division between qualitative and quantitative approaches continues to cut across economics as one of its most heated ongoing fundamental controversies. Most defenders of qualitative approaches tend to reject all mathematical methods and prefer institutional-historical-literary approaches.
Curiously enough, as r increases further to greater than about I 00 order, begins to reemerge. A sequence of period-halving bifurcations happens until at around r = 313 a single stable periodic orbit emerges that then remains as r goes to infinity. 12 drawn from Robbins ( 1979). This represents the case with cr = 10 and ~ = 8/3 that Lorenz studied, but the bifurcation values of r would vary with different values of these parameters, these variations having been intensively studied by Sparrow ( 1982).
Aeroguide 22 - General Dynamics F-111 E/ F by Roger Chesneau