On the Poincaré–Birkhoff theorem as the important result of the theory of dynamical systems

The aim of this work is to study the history of the Poincaré–Birkhoff theorem, which is not only one of the results underlying the theory of dynamical systems, but is important for applications. Until now, the Poincaré–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. Poincaré'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 Poincaré, is the founder of this area of mathematics. Poincaré–Birkhoff theorem is of key importance in understanding the mechanism of the onset of chaotic motion in Hamiltonian systems. The history of the Poincaré–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.