On the fixed points stability for the area-preserving maps

A. P. Markeev

Russian journal of nonlinear dynamics, Tome 11 (2015) no. 3, pp. 503-545.

Résumé

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.

Détail

Keywords: map, canonical transformations, Hamilton system, stability.

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