Advertisement

Generating finite dimensional integrable nonlinear dynamical systems

  • M. Lakshmanan
  • V. K. Chandrasekar
Review Integrable Systems and Solitons

Abstract

In this article, we present a brief overview of some of the recent progress made in identifying and generating finite dimensional integrable nonlinear dynamical systems, exhibiting interesting oscillatory and other solution properties, including quantum aspects. Particularly we concentrate on Lienard type nonlinear oscillators and their generalizations and coupled versions. Specific systems include Mathews-Lakshmanan oscillators, modified Emden equations, isochronous oscillators and generalizations. Nonstandard Lagrangian and Hamiltonian formulations of some of these systems are also briefly touched upon. Nonlocal transformations and linearization aspects are also discussed.

Keywords

Soliton European Physical Journal Special Topic Parameter Symmetry Emden Equation Order ODEs 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983)Google Scholar
  2. 2.
    M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction (John Wiley & Sonc. Inc, New York, 1989)Google Scholar
  3. 3.
    A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations (John Wiley Sons, New York, 1995)Google Scholar
  4. 4.
    S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, (Springer-Verlag, New York, 2003)Google Scholar
  5. 5.
    M. Lakshmanan, S. Rajasekar, Nonlinear Dynamics: Integrability Chaos and Patterns (Springer-Verlag, New York, 2003)Google Scholar
  6. 6.
    F. Calogero, Isochronous Systems (Oxford University Press, Oxford, 2008)Google Scholar
  7. 7.
    G.M. Murphy, Ordinary Differential Equations and their Solutions (Affiliated East-west press, New Delhi, 1969)Google Scholar
  8. 8.
    P.M. Mathews, M. Lakshmanan, Quart. Appl. Math. 32, 215 (1974)MathSciNetzbMATHGoogle Scholar
  9. 9.
    B. Belchev, M.A. Walton, The Morse potential and phase-space quantum mechanics [arXiv.org:1001.4816v1] (2010)
  10. 10.
    D. Zhu, J. Phys. A: Math. Gen. 20, 4331 (1987)ADSCrossRefGoogle Scholar
  11. 11.
    R. Delbourgo, A. Salam, J. Strathdee, Phys. Rev. 187, 19992007 (1969)CrossRefGoogle Scholar
  12. 12.
    R. Koc, M. Koca, J. Phys. A 36, 81058112 (2003)MathSciNetGoogle Scholar
  13. 13.
    R. Gladwin Pradeep, V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, J. Math. Phys. 50, 052901 (2009)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    M. Lakshmanan, K. Eswaran, J. Phys. A 8, 1658 (1975)ADSCrossRefGoogle Scholar
  15. 15.
    P.W. Higgs, J. Phys. A: Math. Gen. 12, 309 (1979)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    H.I. Leemon, J. Phys. A: Math. Gen. 12, 489 (1979)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    J.F. Carinena, M.F. Ranada, M. Santander, Rep. Math. Phys. 54, 285 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    A. Venkatesan, M. Lakshmanan, Phys. Rev. E 55, 5134 (1997)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    J.F. Carinena, M.F. Ranada, M. Santander, M. Senthilvelan, Nonlinearity 17, 1941 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    J.F. Carinena, M.F. Ranada, M. Santander, Ann. Phys. 322, 434 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    A. Tewari, S.N. Pandey, M. Senthilvelan, M. Lakshmanan, [arXiv:1302.0350] (2013)
  22. 22.
    A. Bhuvaneswari, V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, J. Math. Phys. 53, 073504 (2012)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    B. Bagchi, S. Das, S. Ghosh, S. Poria, J. Phys. A 46, 032001 (2013)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    S.C. Cruz, O. Rosas-Ortiz, Dynamical Equations, SIGMA 9, 004 (2013)Google Scholar
  25. 25.
    S.N. Pandey, P.S. Bindu, M. Senthilvelan, M. Lakshmanan, J. Math. Phys. 50, 082702 (2009)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    S.N. Pandey, P.S. Bindu, M. Senthilvelan, M. Lakshmanan, J. Math. Phys. 50, 102701 (2009)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    F.M. Mahomed, P.G.L. Leach, Quaestiones Math. 12, 121 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    P.G.L. Leach, M.R. Feix, S. Bouquet, J. Math. Phys. 29, 2563 (1988)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, Phys. Rev. E 72, 066203 (2005)ADSCrossRefGoogle Scholar
  30. 30.
    V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, Proc. Roy. Soc. London A 461, 2451 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    O. von Roos, Phys. Rev. B 27, 7547 (1983)ADSCrossRefGoogle Scholar
  32. 32.
    V. Chithika Ruby, M. Senthilvelan, M. Lakshmanan, J. Phys. A: Math. Theor. 45, 382002 (2012)ADSCrossRefGoogle Scholar
  33. 33.
    C.M. Bender, D.W. Hook, J. Phys. A: Math. Theor. 44, 372001 (2011)MathSciNetCrossRefGoogle Scholar
  34. 34.
    C.M. Bender, Rep. Prog. Phys. 70, 947 (2007)ADSCrossRefGoogle Scholar
  35. 35.
    M.R. Feix, C. Geronimi, L. Cairo, P.G.L. Leach, R.L. Lemmer, S. Bouquet, J. Phys. A: Math. Gen. 30, 7437 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. 36.
    V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, J. Phys. A: Math. Gen. 40, 4717 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 37.
    F. Calogero, F. Leyvraz, J. Phys. A: Math. Theor. 40, 12931 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  38. 38.
    F. Calogero, F. Leyvraz, J. Phys. A: Math. Theor. 41, 175202 (2008)MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    E.C.G. Sudarshan, N. Mukunda, Classical Dynamics: A Modern Perspective (John Wiley & Sons, New York, 1974)Google Scholar
  40. 40.
    Z.E. Musielak, Standard, J. Phys. A: Math. Theor. 41, 055205 (2008)MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    J.L. Cieśliński, T. Nikiciuk, J. Phys. A: Math. Theor. 43, 175205 (2010)ADSCrossRefGoogle Scholar
  42. 42.
    R. Gladwin Pradeep, V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, J. Phys. A Math. Theor. 42, 135206 (2009)MathSciNetADSCrossRefGoogle Scholar
  43. 43.
    B. Gambier, Acta Math. 33, 1 (1910)MathSciNetCrossRefGoogle Scholar
  44. 44.
    B. Grammaticos, A. Ramani, Phys. A 223, 125 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    B. Grammaticos, A. Ramani, S. Lafortune, Phys. A 253, 260 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    A. Durga Devi, R Gladwin Pradeep, V.K. Chandrasekar, M. Lakshmanan, J. Nonlinear Math. Phys. [arXiv:1207.4611] (accepted) (2013)
  47. 47.
    P. Guha, A.G. Choudhury, B. Grammaticos, SIGMA 7, 028 (2011)Google Scholar
  48. 48.
    R. Gladwin Pradeep, V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, J. Math. Phys. 51, 103513 (2010)MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    J. Chazy, Acta Math. 34, 317 (1911)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    D.L. Gonzalez, O. Piro, Phys. Rev. Lett. 50, 870 (1983)MathSciNetADSCrossRefGoogle Scholar
  51. 51.
    D.L. Gonzalez, O. Piro, Phys. Rev. A 30, 2788 (1984)ADSCrossRefGoogle Scholar
  52. 52.
    M. Hénon, C. Heiles, Astron. J. 69, 73 (1964)ADSCrossRefGoogle Scholar
  53. 53.
    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, Products (Academic press, London, 1980)Google Scholar
  54. 54.
    M. Lakshmanan, R. Sahadevan, Phys. Rep. 224, 1 (1993)MathSciNetADSCrossRefGoogle Scholar
  55. 55.
    R. Conte, M. Musette, C. Verhoeven 144, 888 (2005)MathSciNetzbMATHGoogle Scholar
  56. 56.
    T. Bountis, H. Segur, F. Vivaldi, Phys. Rev. A 25, 1257 (1982)MathSciNetADSCrossRefGoogle Scholar
  57. 57.
    P. Painlevé, C. R. Acad. Sc. Paris 143, 1111 (1906)Google Scholar
  58. 58.
    E.L. Ince, Ordinary Differential Equations (Dover, New York, 1956)Google Scholar
  59. 59.
    V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, J. Phys. A: Math. Theor. 39, L69 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  60. 60.
    P. Guha, A.G. Choudhury, Pramana 77, 917 (2011)ADSCrossRefGoogle Scholar
  61. 61.
    G. Darboux, Bull. Sci. Math. 32, 6096, 123144, 151200 (1878)Google Scholar
  62. 62.
    O. Cornejo-Pérez, H.C. Rosu, Prog. Theor. Phys. 114 533 (2005)ADSzbMATHCrossRefGoogle Scholar
  63. 63.
    R. Conte, The Painlevé Property: One Century Later (New York, Springer, 1999)Google Scholar
  64. 64.
    M.A. Reyes, H.C. Rosu, J. Phys. A: Math. Theor. 41, 285206 (2008)MathSciNetCrossRefGoogle Scholar
  65. 65.
    V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, J. Nonlinear Math. Phys. 12, 184 (2005)Google Scholar
  66. 66.
    T. Hazra, V.K. Chandrasekar, R Gladwin Pradeep, M. Lakshmanan, J. Math. Phys. 53, 023511 (2012)MathSciNetADSCrossRefGoogle Scholar
  67. 67.
    J.M. Dixon, J.A. Tuszynski, Phys. Rev. A 41, 416673 (1990)MathSciNetCrossRefGoogle Scholar
  68. 68.
    G. Darboux, Bull. Sci. Math. 32, 6096, 123144, 151200 (1878)Google Scholar
  69. 69.
    O. Cornejo-Pérez, H.C. Rosu, Prog. Theor. Phys. 114 533 (2005)ADSzbMATHCrossRefGoogle Scholar
  70. 70.
    M.A. Reyes, H.C. Rosu, J. Phys. A: Math. Theor. 41, 285206 (2008)MathSciNetCrossRefGoogle Scholar
  71. 71.
    T. Hazra, V.K. Chandrasekar, R Gladwin Pradeep, M. Lakshmanan, J. Math. Phys. 53, 023511 (2012)MathSciNetADSCrossRefGoogle Scholar
  72. 72.
    M. Ablowitz, P. Clarkson, Solitons, Nonlinear Evolution Equations, Inverse Scattering, (Cambridge University Press, Cambridge, 1991)Google Scholar
  73. 73.
    J.M. Dixon, J.A. Tuszynski, Phys. Rev. A 41, 416673 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Centre for Nonlinear Dynamics, School of PhysicsBharathidasan UniversityTiruchirappalliIndia

Personalised recommendations