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A Protogeometric Nature Goddess from Knossos by J.N. Coldstream

By J.N. Coldstream

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In Poincar´e’s paper, he described new mathematical ideas such as homoclinic points. The memoir was about to be published in Acta Mathematica when an error was found by the editor. This error in fact led to further discoveries by Poincar´e, which are now considered to be the beginning of chaos theory. The memoir was published later in 1890. Poincar´e’s research into orbits about Lagrange points and low-energy transfers was not utilized for more than a century afterwards. 8 The interesting thing about Poincar´e’s work was that it did not solve the problem posed.

A. Andronov 11 carried on with the study of nonlinear oscillators and in 1937 introduced together with Lev S. Pontryagin 12 the notion of coarse systems. They were formalizing the understanding garnered from the study of nonlinear oscillators, the understanding that many of the details on how these oscillators work do not affect the overall picture of the phase–space: there will still be limit cycles if one changes the dissipation or spring force function by a little bit. And changing the system a little bit has the great advantage of eliminating exceptional cases in the mathematical analysis.

The operation for comparing orbits to establish their equivalence changes with the different notions of stability. 2. The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes.

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