The Catastrophe Machine
4. Doctor Zeeman's Original Catastrophe Machine
This device, invented by the mathematician Christopher Zeeman (now Sir E. C. Zeeman, K.B., F.R.S.), consists of a wheel which is tethered by an elastic to a fixed point in its plane. The control input to the system is another elastic attached to the same point as the first and roughly of the same length. The other end of the elastic can be moved about an area diametrically opposite to the fixed point. This particular instantiation of the concept is about one meter high. The detail shows the way the two elastics are attached to the wheel.
The catastrophe locus (roughly sketched out in chalk) is entirely contained in a small region of the plane. If the control point is displaced outside that region, the wheel tracks smoothly. 
The wheel responds discontinuously when the control point crosses the catastrophe locus. 


The "sheets" corresponding to the two cusps, and the fold lines which issue from the cusps, are connected as shown on the right. Here the stippled areas correspond to local minima; the unstippled ones are inaccessible. The sheets all extend across the picture  they have been cut away for visibility's sake.



What is a mathematical catastrophe?

An algebraic version of the double well

The Cusp Catastrophe

Doctor Zeeman's Original Catastrophe Machine

The Catastrophe Machine's unperturbed potential function is y=x^{4}

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