Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2011, Article ID 831647, 14 pages
http://dx.doi.org/10.1155/2011/831647
Research Article

First Integrals for Two Linearly Coupled Nonlinear Duffing Oscillators

1Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan
2Department of Mathematics, LUMS, School of Science and Engineering, Lahore Cantt 54792, Pakistan
3Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

Received 6 September 2010; Revised 1 December 2010; Accepted 3 January 2011

Academic Editor: G. Rega

Copyright © 2011 R. Naz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. F. Dimentberg and A. S. Bratus, “Bounded parametric control of random vibrations,” Proceedings of the Royal Society. A, vol. 456, no. 2002, pp. 2351–2363, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. V. N. Pilipchuk, “Transitions from strongly to weakly-nonlinear dynamics in a class of exactly solvable oscillators and nonlinear beat phenomena,” Nonlinear Dynamics, vol. 52, no. 3, pp. 263–276, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. E. Noether, “Invariant Variationsprobleme,” Nachrichten von der Königlicher Gesellschaft den Wissenschaft zu Göttingen, Mathematisch-Physikalische Klasse Heft, vol. 2, pp. 235–257, 1918, English translation in Transport Theory and Statical Physics, vol. 1, pp. 186–207, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1986.
  5. J. Douglas, “Solution of the inverse problem of the calculus of variations,” Transactions of the American Mathematical Society, vol. 50, pp. 71–128, 1941. View at Google Scholar · View at Zentralblatt MATH
  6. J. Hietarinta, Direct Methods for the Search of the Second Invariant, Report Series, Department of Physical Sciences, University of Turku, Turku, Finland, 1986.
  7. H. R. Lewis and P. G. L. Leach, “A direct approach to finding exact invariants for one-dimensional time-dependent classical Hamiltonians,” Journal of Mathematical Physics, vol. 23, no. 12, pp. 2371–2374, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. H. Steudel, “Über die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssätzen,” Zeitschrift für Naturforschung A, vol. 17, pp. 129–132, 1962. View at Google Scholar
  9. S. Moyo and P. G. L. Leach, “Symmetry properties of autonomous integrating factors,” SIGMA, vol. 1, 12 pages, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. H. Kara and F. M. Mahomed, “Noether-type symmetries and conservation laws via partial Lagrangians,” Nonlinear Dynamics, vol. 45, no. 3-4, pp. 367–383, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. A. H. Kara, F. M. Mahomed, I. Naeem, and C. Wafo Soh, “Partial Noether operators and first integrals via partial Lagrangians,” Mathematical Methods in the Applied Sciences, vol. 30, no. 16, pp. 2079–2089, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. P. A. Damianou and C. Sophocleous, “Classification of Noether symmetries for Lagrangians with three degrees of freedom,” Nonlinear Dynamics, vol. 36, no. 1, pp. 3–18, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. I. Naeem and F. M. Mahomed, “Noether, partial Noether operators and first integrals for a linear system,” Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 70–82, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. I. Naeem and F. M. Mahomed, “Partial Noether operators and first integrals for a system with two degrees of freedom,” Journal of Nonlinear Mathematical Physics, vol. 15, supplement 1, pp. 165–178, 2008. View at Publisher · View at Google Scholar
  15. W. T. van Horssen, “A perturbation method based on integrating vectors and multiple scales,” SIAM Journal on Applied Mathematics, vol. 59, no. 4, pp. 1444–1467, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. S. B. Waluya and W. T. van Horssen, “On approximations of first integrals for a strongly nonlinear forced oscillator,” Nonlinear Dynamics, vol. 33, no. 3, pp. 225–252, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, Berlin, Germany, 2nd edition, 1996.
  18. V. M. Gorringe and P. G. L. Leach, “Lie point symmetries for systems of second order linear ordinary differential equations,” Quaestiones Mathematicae, vol. 11, no. 1, pp. 95–117, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. C. Wafo Soh and F. M. Mahomed, “Linearization criteria for a system of second-order ordinary differential equations,” International Journal of Non-Linear Mechanics, vol. 36, no. 4, pp. 671–677, 2001. View at Publisher · View at Google Scholar
  20. C. Wafo Soh and F. M. Mahomed, “Canonical forms for systems of two second-order ordinary differential equations,” Journal of Physics A, vol. 34, no. 13, pp. 2883–2911, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH