Bifurcations and catastrophe theory

The solution of the cubic equation, and its associated cusp discriminant, has an interesting history itself. Guedj (2000) provides a fictionalized account of this story in a delightful book. Catastrophe theory, popularized in the 1970s by Rene Thom and Chris Zeeman, entered our language again from the French word for sudden change ("catastrophe"), and in the mid 1970s the cusp catastrophe attracted lots of mathematicians and physicists who saw ways of explaining many kinds of biological and social phenomena without having to know the details of the application (Kolata 1977). An example of relaxation oscillation in a marine system is given by James et a/. (2003); one of chaos by Chattopadhyay and Sarkar (2003). The next bifurcation in the one dimensional series is called the swallowtail (the names in use are still the ones picked by Rene Thom (1972/1975)) and corresponds to the steady states of the differential equation dx/dt =— x4 + ax2 + ^x + 7. Hernandez and Barradas (2003) put a nice ecological context around bifurcations and catastrophes. Readers of this book interested in conservation biology will surely already know what must be the simplest of the bifurcations (so simple that it is never even named, but see below), which occurs in the Levins patch model (Levins 1969, 1970). In this model, one focuses on the dynamics of the fraction of occupied patches in a metapopulation connected by dispersal. Patches become extinct at a rate m and are colonized in proportion to the number of occupied patches with proportionality constant c. The dynamics thus become

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