Geodesic
On the fixed points stability for the area-preserving maps
Russian journal of nonlinear dynamics, Tome 11 (2015) no. 3, pp. 503-545

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We study area-preserving maps. The map is assumed to have a fixed point and be analytic in its small neighborhood. The main result is a developed constructive algorithm for studying the stability of the fixed point in critical cases when members of the first degrees (up to the third degree inclusive) in a series specifying the map do not solve the issue of stability. As an application, the stability problem is solved for a vertical periodic motion of a ball in the presence of impacts with an ellipsoidal absolutely smooth cylindrical surface with a horizontal generatrix. Study of area-preserving maps originates in the Poincaré section surfaces method [1]. The classical works by Birkhoff [2–4], Levi-Civita [5], Siegel [6, 7], Moser [7–9] are devoted to fundamental aspects of this problem. Further consideration of the objectives is contained in the works by Russman [10], Sternberg [11], Bruno [12, 13], Belitsky [14] and other authors.
Keywords: map, canonical transformations, Hamilton system, stability. 
@article{ND_2015_11_3_a4,
     author = {A. P. Markeev},
     title = {On the fixed points stability for the area-preserving maps},
     journal = {Russian journal of nonlinear dynamics},
     pages = {503--545},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2015_11_3_a4/}
}
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A. P. Markeev. On the fixed points stability for the area-preserving maps. Russian journal of nonlinear dynamics, Tome 11 (2015) no. 3, pp. 503-545. http://geodesic.mathdoc.fr/item/ND_2015_11_3_a4/