Model of Poincaré's Grenzcycle by Richard P. Baker, Baker #517

This model is one of several hundred designed by Richard P. Baker, a mathematics faculty member at the University of Iowa. It has a black wooden base with a plaster surface atop it. The sides of the plaster are painted black, the top is white. A typed paper tag attached to underside of the base reads: No. 517 (/) Grenzcycle. The term “Grenzcycle” translates from German into English as “limit cycle.”

In an 1882 paper, the French mathematician Henri Poincaré introduced the concept of a limit cycle. According to his definition, the limit cycle is a closed curve that satisfied a differential equation which other closed curves satisfying the same equation approached asymptotically. In this model, which follows Poincaré’s example, the limit cycle is a circle, with one spiral approaching it from the outside and a second approaching it from the inside.

References:

See the article on limit cycles at http://www.scholarpedia.org/ , accessed August 12, 2020.

Richard P. Baker, Mathematical Models, Iowa City, 1931, p. 7. This source gives the equations Baker sought to graph.as: 𝑧= 𝜃 + log𝜌 −1/2 log(1− 𝜌²), from the discussion of the differential equation : 𝑥+𝑦𝑦′=(𝑥𝑦′−𝑦)(𝑥²+𝑦² −1).

H. Poincaré, “Sur les courbes définies par une équation différentielle,” Journal de mathématiques pures et appliquées, 1882, (III) 8, pp. 251-296, esp. p. 280.

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