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 Nelin. Dinam., 2015, Volume 11, Number 3, Pages 503–545 (Mi nd493)

Original papers

On the fixed points stability for the area-preserving maps

A. P. Markeev

A.Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526, Russia

Abstract: 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

 Funding Agency Grant Number Russian Foundation for Basic Research 14.01.00380 Ministry of Education and Science of the Russian Federation ÍØ-2363.2014.1

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UDC: 531.36
MSC: 70H05, 70H15, 70E50
Revised: 15.09.2015

Citation: A. P. Markeev, “On the fixed points stability for the area-preserving maps”, Nelin. Dinam., 11:3 (2015), 503–545

Citation in format AMSBIB
\Bibitem{Mar15} \by A.~P.~Markeev \paper On the fixed points stability for the area-preserving maps \jour Nelin. Dinam. \yr 2015 \vol 11 \issue 3 \pages 503--545 \mathnet{http://mi.mathnet.ru/nd493} 

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This publication is cited in the following articles:
1. 62, no. 4, 2017, 228–232
2. A. P. Markeev, “Ob ustoichivosti permanentnykh vraschenii diska pri nalichii ego soudarenii s gorizontalnoi ploskostyu”, Nelineinaya dinam., 11:4 (2015), 685–707
3. Boris S. Bardin, Victor Lanchares, “On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy”, Regul. Chaotic Dyn., 20:6 (2015), 627–648
4. A. P. Markeev, “Ob ustoichivosti dvukhzvennoi traektorii paraboloidnogo bilyarda Birkgofa”, Nelineinaya dinam., 12:1 (2016), 75–90
5. A. P. Markeev, “On the stability of periodic trajectories of a planar Birkhoff billiard”, Proc. Steklov Inst. Math., 295 (2016), 190–201
6. A. P. Markeev, “Stability in a Case of Motion of a Paraboloid Over a Plane”, Mech. Sol., 51:6 (2016), 623–631
7. A. P. Markeev, “The stability of two-link trajectories of a Birkhoff billiard”, J. Appl. Math. Mech., 80:4 (2016), 280–289
8. A. P. Markeev, “Ob ustoichivosti dvizheniya mayatnika Maksvella”, Nelineinaya dinam., 13:2 (2017), 207–226
9. Rodrigo Gutierrez, Claudio Vidal, “Stability of Equilibrium Points for a Hamiltonian Systems with One Degree of Freedom in One Degenerate Case”, Regul. Chaotic Dyn., 22:7 (2017), 880–892
10. F. L. Chernous'ko, “Translational motion of a chain of bodies in a resistive medium”, J. Appl. Math. Mech., 81:4 (2017), 256–269
11. A. P. Markeev, “On the stability of the regular Grioli precession in a particular case”, Mech. Sol., 53:2 (2018), S1–S14
12. A. P. Markeev, “The stability of relative equilibrium positions of a pendulum on a mobile platform”, Dokl. Phys., 63:10 (2018), 441–445
13. A. P. Markeev, “Stability in the regular precession of an asymmetrical gyroscope in the critical case of fourth-order resonance”, Dokl. Phys., 63:7 (2018), 297–301
14. B. S. Bardin, “On the stability of a periodic Hamiltonian system with one degree of freedom in a transcendental case”, Dokl. Math., 97:2 (2018), 161–163
15. Anatoly P. Markeev, “On the Stability of the Regular Precession of an Asymmetric Gyroscope at a Second-order Resonance”, Regul. Chaotic Dyn., 24:5 (2019), 502–510
16. Boris S. Bardin, Víctor Lanchares, “Stability of a One-degree-of-freedom Canonical System in the Case of Zero Quadratic and Cubic Part of a Hamiltonian”, Regul. Chaotic Dyn., 25:3 (2020), 237–249
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