On the fixed points stability for the area-preserving maps

    2015, Vol. 11, No. 3, pp.  503-545

    Author(s): Markeev A. P.

    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
    Citation: Markeev A. P., On the fixed points stability for the area-preserving maps, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 3, pp.  503-545

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