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# Recipe for catastrophe

Last year we published a paper titled “Electromechanical Catastrophe”. I am planning to give a lecture on this topic in my course on Advanced Elasticity. The mathematics of catastrophe is old, but can be used more widely. In describing instability of a system subject to a single load, we plot a two-dimensional diagram, called the bifurcation diagram, with one axis representing values of the load, and another axis representing states of the system. The bifurcation diagram displays branches of states of equilibrium as the load varies.

What do we do if a system is subject to two independent loads? The introduction of the paper gives a generic description of the mathematics. The rest of the paper describes an example using a combination of calculation and experiment. The idea of catastrophe is old, simple, and general. It deserves our attention.

An everyday example of two-load system is a fixed amount of water. The two loads are pressure and temperature. A proxy for the state of the system can be the volume of water. The van der Waals model gives a surface in the three-dimensional space of pressure, temperature, and volume. The surface in the three-dimensional space is smooth, but its projection onto the pressure-temperature plane (the loading plane) is not. On the pressure-temperature plane, the critical point is a cusp. Typically, water does not reach the folds. Instead, liquid and vapor coexist. The pressure-temperature diagram is the phase diagram, which tells us about boiling, dews, and pressure cookers. Thus, the liquid-gas transition of water is a thermomechanical catastrophe. We live in catastrophe. We are catastrophe.

Fundamental ideas generic to catastrophe of a two-load system include cusp (critical point), fold, snap forward, snap backward, hysteresis, coexistent states.

Here is a recipe to cook up a catastrophe. If you have a one-load system that suffers a snap-through instability, you can introduce another control variable, and make the system into a system of two control variables. The control variable can be another load, or just a geometric parameter that you can vary. Now you have your own catastrophe.

It should be easy to create thermochemical catastrophe, thermoelectrical catastrophe, chemomechanical catastrophe, electrochemical catastrophe...

Attachment | Size |
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363 Electromechanical Catastrophe.pdf | 692.14 KB |

1976 zeeman catastrophe theory.pdf | 1.61 MB |

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## Comments

## Zeeman's Christman Lecture 1978 on cacastrophe theory

Here is the link to the video of his lecture. I have also uploaded his Scientific American article.

## Such insight

Dear Zhigang,

Thanks for sharing the link. Brilliant insights by Professor Zeeman.

Shailendra