One-parameter families of brake orbits in dynamical systems

Lennard Bakker

Colloquium Mathematicae (1999)

  • Volume: 82, Issue: 2, page 201-217
  • ISSN: 0010-1354

Abstract

top
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.

How to cite

top

Bakker, Lennard. "One-parameter families of brake orbits in dynamical systems." Colloquium Mathematicae 82.2 (1999): 201-217. <http://eudml.org/doc/210757>.

@article{Bakker1999,
abstract = {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.},
author = {Bakker, Lennard},
journal = {Colloquium Mathematicae},
keywords = {nonequilibrium solution; brake orbit; periodic brake orbits},
language = {eng},
number = {2},
pages = {201-217},
title = {One-parameter families of brake orbits in dynamical systems},
url = {http://eudml.org/doc/210757},
volume = {82},
year = {1999},
}

TY - JOUR
AU - Bakker, Lennard
TI - One-parameter families of brake orbits in dynamical systems
JO - Colloquium Mathematicae
PY - 1999
VL - 82
IS - 2
SP - 201
EP - 217
AB - 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.
LA - eng
KW - nonequilibrium solution; brake orbit; periodic brake orbits
UR - http://eudml.org/doc/210757
ER -

References

top
  1. [1] A. Ambrosetti, V. Benci and V. Long, A note on the existence of multiple brake orbits, Nonlinear Anal. 21 (1993), 643-649. Zbl0811.70015
  2. [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. [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. [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. Zbl0674.34034
  5. [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. [6] B. Buffoni and F. Giannoni, Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems, Discrete Contin. Dynam. Systems. 1 (1995), 217-222. Zbl0868.34032
  7. [7] C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987), 310-321. Zbl0622.34033
  8. [8] C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 311 2 (1989), 433-486. 
  9. [9] V. Coti Zelati and E. Serra, Multiple brake orbits for some classes of singular Hamiltonian systems, Nonlinear Anal. 20 (1993), 1001-1012. Zbl0778.34026
  10. [10] R. L. Devaney, Reversible Diffeomorphisms and Flows, Trans. Amer. Math. Soc. 218 (1976), 89-113. Zbl0363.58003
  11. [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. Zbl0546.58040
  12. [11] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer, New York, 1973. Zbl0294.58004
  13. [24] E. van Groesen, Duality between period and energy of certain periodic Hamiltonian motions, J. London Math. Soc. (2) 34 (1986), 435-448. Zbl0582.34052
  14. [25] E. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy, J. Math. Anal. Appl. 132 (1988), 1-12. Zbl0665.70022
  15. [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. Zbl0569.70017
  16. [14] K. Meyer, Hamiltonian systems with a finite symmetry, J. Differential Equations 41 (1981), 228-238. Zbl0438.70022
  17. [15] K. Meyer and G. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer, New York, 1992. Zbl0743.70006
  18. [16] D. C. Offin, A class of periodic orbits in classical mechanics, J. Differential Equations 66 (1987), 90-117. Zbl0616.34040
  19. [17] P. A. Rabinowitz, On a theorem of Weinstein, ibid. 68 (1987), 332-343. 
  20. [18] P. A. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear Anal. 11 (1987), 599-611. 
  21. [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. Zbl0789.58040
  22. [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. 
  23. [21] H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z. 51 (1949), 197-216. Zbl0030.22103
  24. [22] C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, reprint of 1971 edition, Springer, New York, 1995. 
  25. [23] A. Szulkin, An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems, Math. Ann. 283 (1989), 241-255. Zbl0642.58030
  26. [26] A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), 507-518. Zbl0403.58001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.