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1999 | 82 | 2 | 201-217
Tytuł artykułu

One-parameter families of brake orbits in dynamical systems

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We give a clear and systematic exposition of one-parameter families of brake orbits in dynamical systems on product vector bundles (where the fiber has the same dimension as the base manifold). A generalized definition of a brake orbit is given, and the relationship between brake orbits and periodic orbits is discussed. The brake equation, which implicitly encodes information about the brake orbits of a dynamical system, is defined. Using the brake equation, a one-parameter family of brake orbits is defined as well as two notions of nondegeneracy by which a given brake orbit embeds into a one-parameter family of brake orbits. The duality between the two notions of nondegeneracy for a brake orbit in a one-parameter family is described. Finally, four ways in which a given periodic brake orbit generates a one-parameter family of periodic brake orbits are detailed.
Słowa kluczowe
Rocznik
Tom
82
Numer
2
Strony
201-217
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-11-16
poprawiono
1999-06-18
Twórcy
  • Department of Mathematics, Brigham Young University, 292 Talmage Math/Computer Building, PO Box 26539, Provo, UT 84602-6539, U.S.A.
Bibliografia
  • [1] A. Ambrosetti, V. Benci and V. Long, A note on the existence of multiple brake orbits, Nonlinear Anal. 21 (1993), 643-649.
  • [2] L. F. Bakker, An existence theorem for periodic brake orbits and heteroclinic connections, preprint 43, Department of Mathematics, Univ. of Nevada, Reno, 1999.
  • [3] L. F. Bakker, Brake orbits and magnetic twistings in two degrees of freedom Hamiltonian dynamical systems, Ph.D. thesis, Queen's Univ., Kingston, Canada, 1997.
  • [4] V. Benci and F. Giannoni, A new proof of the existence of a brake orbit, in: Advanced Topics in the Theory of Dynamical Systems, Academic Press, Boston, 1989, 37-49.
  • [5] G. D. Birkhoff, The restricted problem of three bodies, reprinted from Rend. Circ. Mat. Palermo 39 (1915), in: George David Birkhoff, Collected Mathematical Papers, Vol. 1, Amer. Math. Soc., New York, 1950, 682-751.
  • [6] B. Buffoni and F. Giannoni, Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems, Discrete Contin. Dynam. Systems. 1 (1995), 217-222.
  • [7] C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987), 310-321.
  • [8] C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 311 2 (1989), 433-486.
  • [9] V. Coti Zelati and E. Serra, Multiple brake orbits for some classes of singular Hamiltonian systems, Nonlinear Anal. 20 (1993), 1001-1012.
  • [10] R. L. Devaney, Reversible Diffeomorphisms and Flows, Trans. Amer. Math. Soc. 218 (1976), 89-113.
  • [12] H. Gluck and W. Ziller, Existence of periodic motions of conservative systems, in: Seminar on Minimal Submanifolds, Princeton Univ. Press, Princeton, 1983, 65-98.
  • [11] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer, New York, 1973.
  • [24] E. van Groesen, Duality between period and energy of certain periodic Hamiltonian motions, J. London Math. Soc. (2) 34 (1986), 435-448.
  • [25] E. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy, J. Math. Anal. Appl. 132 (1988), 1-12.
  • [13] H. Hofer and J. F. Toland, Homoclinic, heteroclinic, and periodic orbits for a class of indefinite Hamiltonian systems, Math. Ann. 268 (1984), 387-403.
  • [14] K. Meyer, Hamiltonian systems with a finite symmetry, J. Differential Equations 41 (1981), 228-238.
  • [15] K. Meyer and G. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer, New York, 1992.
  • [16] D. C. Offin, A class of periodic orbits in classical mechanics, J. Differential Equations 66 (1987), 90-117.
  • [17] P. A. Rabinowitz, On a theorem of Weinstein, ibid. 68 (1987), 332-343.
  • [18] P. A. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear Anal. 11 (1987), 599-611.
  • [19] P. A. Rabinowitz, Some recent results on heteroclinic and other connecting orbits in Hamiltonian systems, in: Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations, Pitman Res. Notes Math. Ser. 243, Longman Sci. Tech., 1992, 157-168.
  • [20] O. R. Ruiz M., Existence of brake orbits in Finsler mechanical systems, in: Geometry and Topology, Lecture Notes in Math. 597, Springer, Berlin, 1977, 542-567.
  • [21] H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z. 51 (1949), 197-216.
  • [22] C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, reprint of 1971 edition, Springer, New York, 1995.
  • [23] A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann. 283 (1989), 241-255.
  • [26] A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), 507-518.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv82i2p201bwm
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