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International Journal of Theoretical Physics

, Volume 31, Issue 2, pp 221–228 | Cite as

Forced periodic oscillations and the Jones polynomial

  • Giuseppe Gaeta
Article

Abstract

We show that forced periodic oscillations in a nonlinear damped oscillator can be classified by means of the Jones (knot) polynomial; this is done by associating to any periodic oscillation a braid. We also discuss the relation of this approach to (Lie-point) symmetry analysis of the associated differential equations.

Keywords

Differential Equation Field Theory Elementary Particle Quantum Field Theory Periodic Oscillation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Alexander, J. W. (1928).Transactions of the American Mathematical Society,20, 275–306.Google Scholar
  2. Atiyah, M. (1990).The Geometry and Physics of Knots, Cambridge University Press, Cambridge.Google Scholar
  3. Birman, J., and Williams, R. F. (1983a).Topology,22, 47–82.Google Scholar
  4. Birman, J., and Williams, R. F. (1983b).Contemporary Mathematics,20, 1–60.Google Scholar
  5. Bluman, G. W., and Kumei, S. (1989).Symmetries and Differential Equations, Springer, Berlin.Google Scholar
  6. Burde, G., and Zieschang, H. (1986).Knots, de Gruyter.Google Scholar
  7. Champagne, B., Hereman, W., and Winternitz, P. (1990). The computer calculation of Lie point symmetries of large systems of differential equations, Preprint CRM-1689.Google Scholar
  8. Cicogna, G. (1990).Journal of Physics A,23, L1339-L1343.Google Scholar
  9. Cicogna, G., and Gaeta, G. (1990). Lie-point symmetries and bifurcation theory, preprint CPT Polytechnique.Google Scholar
  10. Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millet, K. C., and Ocneanu, A. (1985). [HOMFLY]:Bulletin of the American Mathematical Society,12, 239–246.Google Scholar
  11. Frohlich, J., and King, C. (1989).International Journal of Modern Physics A,4, 5321.Google Scholar
  12. Gaeta, G. (1990). Lie-point symmetries and periodic solutions for autonomous ODE, preprint CPT Polytechnique.Google Scholar
  13. Gaeta, G. (1991). Bifurcation theory and nonlinear symmetries,Nonlinear Analysis, to appear.Google Scholar
  14. Holmes, P. J. (1986).Physica D,21, 7–41.Google Scholar
  15. Holmes, P. J. (1988). Knots and orbit genealogies in nonlinear oscillators, inNew Directions in Dynamical Systems, T. Bedford and J. Swift, eds., Cambridge University Press, Cambridge.Google Scholar
  16. Holmes, P. J., and Williams, R. F. (1985).Archives for Rational Mechanics and Analysis,90, 115–194.Google Scholar
  17. Jimbo, K., ed. (1989). Yang-Baxter equation in integrable systems, World Scientific, Singapore.Google Scholar
  18. Jones, V. F. R. (1985).Bulletin of the American Mathematical Society,12, 103–112.Google Scholar
  19. Jones, V. F. R. (1986).Notices of the American Mathematical Society,33, 219–225.Google Scholar
  20. Jones, V. F. R. (1987).Annals of Mathematics,126, 335–388.Google Scholar
  21. Kauffman, L. H. (1983).Formal Knot Theory, Princeton University Press, Princeton, New Jersey.Google Scholar
  22. Kauffman, L. H. (1987).On Knots, Princeton University Press, Princeton, New Jersey.Google Scholar
  23. Kauffman, L. H. (1988).American Mathematical Monthly,95, 195–242.Google Scholar
  24. Kauffman, L. H. (1989). Polynomial invariants in knot theory, inBraid Group, Knot Theory and Statistical Mechanics, C. N. Yang and M. L. Ge, eds., World Scientific, Singapore.Google Scholar
  25. Kauffman, L. H. (1990).L'Enseignement Mathematique,36, 1.Google Scholar
  26. Kauffman, L. H. (to appear). Statistical mechanics and the Jones polynomial, in Proceedings of the Artin Braid Group Conference, A.M.S.,Contemporary Mathematics, to appear.Google Scholar
  27. Lusanna, L., ed. (1990).Knots, Topology and Quantum Field Theories, World Scientific, Singapore.Google Scholar
  28. Mielke, A. (1990). Topological methods for discrete dynamical systems,GAMM-Mitt. 2, 19–37.Google Scholar
  29. Olver, P. J. (1986).Applications of Lie Groups to Differential Equations, Springer, Berlin.Google Scholar
  30. Ovsjannikov, L. V. (1982).Group Properties of Differential Equations, Academic Press.Google Scholar
  31. Reidemeister, K. (1948).Knotentheorie, Chelsea, New York.Google Scholar
  32. Rolfsen, D. (1976).Knots and Links, Publish or Perish.Google Scholar
  33. Wadati, M., Deguchi, T., and Akutsu, Y. (1989).Physics Reports,180, 247–332.Google Scholar
  34. Winternitz, P. (1990). Group theory and exact solutions of partially integrable differential systems, inPartially Integrable Evolution Equations in Physics, R. Conte and N. Boccara, eds., Kluwer.Google Scholar
  35. Witten, E. (1989a).Communications in Mathematical Physics,121, 351.Google Scholar
  36. Witten, E. (1989b).Nuclear Physics B,322, 629.Google Scholar
  37. Yang, C. N., and Ge, M. L., eds. (1989).Braid Group, Knot Theory and Statistical Mechanics, World Scientific, Singapore.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Giuseppe Gaeta
    • 1
  1. 1.Centre de Physique ThéoriqueEcole PolytechniquePalaiseauFrance

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