On the Poincar´e-Birkhoff theorem as the important result of the theory of dynamical systems
https://doi.org/10.22405/2226-8383-2022-23-1-209-222
Abstract
The aim of this work is to study the history of the Poincar´e-Birkhoff theorem, which is not only one of the results underlying the theory of dynamical systems, but is important for applications. Until now, the Poincar´e-Birkhoff theorem has been considered historically only fragmentarily and has not been the subject of consistent historical research. The research is based on the analysis of original works, historical and scientific literature, involving the recollections of participants in the described events. Poincar´e’s idea was to establish the periodic motions of dynamical systems using the geometric theorem he proposed. Periodic movements, in turn, were supposed to serve as a basis for studying other, complex movements. The search for a proof was a powerful impetus for Birkhoff in the construction of the theory of dynamical systems, who, together with Poincar´e, is the founder of this area of mathematics. Poincar´e- Birkhoff theorem is of key importance in understanding the mechanism of the onset of chaotic motion in Hamiltonian systems. The history of the Poincar´e-Birkhoff theorem is not complete;
it plays a significant role in the modern theory of dynamical systems and its applications.
The search continues for a proof of a multidimensional analogue of the theorem, its various generalizations, and further applications.
Keywords
integration of differential equations,
three-body problem,
dynamical system,
periodic motions,
chaotic motions.
Supporting agencies: Thе work was supported by the RFBR, project No. 20-011-00402 A
About the Author
Ravil’ Rafkatovich Mukhin
Stary Oskol Technological Institute of National Research University of Science and Technology “MISiS”
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
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For citations:
Mukhin R.R. On the Poincar´e-Birkhoff theorem as the important result of the theory of dynamical systems. Chebyshevskii Sbornik. 2022;23(1):209-222. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-1-209-222
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