On the Symmetries of a Liénard Type Nonlinear Oscillator Equation

  • R. Mohanasubha
  • V. K. Chandrasekar
  • M. Senthilvelan
  • M. LakshmananEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 266)


In the contemporary nonlinear dynamics literature, the nonlinear oscillator equation \(\ddot{x}+kx \dot{x}+\frac{k^2}{9}x^3+\tilde{\lambda } x=0\) is being analyzed in various contexts both classically and quantum mechanically. Classically this nonlinear oscillator equation has been shown to admit three different types of dynamics depending upon the sign and magnitude of the parameter \(\tilde{\lambda }\), namely (i) \(\tilde{\lambda }=0\), (ii) \(\tilde{\lambda }>0\) and (iii) \(\tilde{\lambda }<0\). By considering its importance, in this paper, we present the symmetries of its Lagrangian and underlying equation of motion for all the three cases. In particular, we present Lie point symmetries, \(\lambda \)-symmetries, Noether symmetries and telescopic symmetries of this equation. The utility of the symmetries for all the three cases is demonstrated explicitly.


Nonlinear oscillators Lie point symmetries \(\lambda \)-symmetries Noether symmetries Telescopic vector fields 



RMS acknowledges the University Grants Commission (UGC), New Delhi, India, for the award of a Dr. D. S. Kothari Post Doctoral Fellowship [Award No. F.4-2/2006 (BSR)/PH/17-18/0004]. The work of M.S. forms part of CSIR research project under Grant No. 03(1397)/17/EMR-II. The work of V.K.C. was supported by the SERB-DST Fast Track scheme for young scientists under Grant No. YSS/2014/000175. The work of M.L. is supported by a DST- SERB Distinguished Fellowship program.


  1. 1.
    Bhuvaneswari, A., Kraenkel, R.A., Senthilvelan, M.: Nonlinear Anal. Real World Appl. 13, 1102–1114 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bluman, G.W., Anco, S.C.: Symmetries and Integration Methods for Differential Equations. Springer, New York (2002)zbMATHGoogle Scholar
  3. 3.
    Chandrasekhar, S.: An Introduction to the Study of Stellar Structure. Dover, New York (1957)zbMATHGoogle Scholar
  4. 4.
    Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: Phys. Rev. E 72, 066203 (2005)CrossRefGoogle Scholar
  5. 5.
    Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: Proc. R. Soc. Lond. A 461, 2451–2476 (2005)CrossRefGoogle Scholar
  6. 6.
    Chandrasekar, V.K., Pandey, S.N., Senthilvelan, M., Lakshmanan, M.: Chaos, Solitons Fractals 26, 1399–1406 (2005)Google Scholar
  7. 7.
    Chandrasekar, V.K., Pandey, S.N., Senthilvelan, M., Lakshmanan, M.: J. Phys. A Math. Gen. 39, L69–L76 (2006)CrossRefGoogle Scholar
  8. 8.
    Chithiika Ruby, V., Senthilvelan, M., Lakshmanan, M.: J. Phys. A Math. Theor. 45, 382002 (2012)CrossRefGoogle Scholar
  9. 9.
    Choudhuri, A.: On the symmetries of the modified Emden-type equation. arXiv:1304.5826v1
  10. 10.
    Gaeta, G.: J. Nonlinear Math. Phys. 16, 107 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gelfand, I.M., Fomin, S.V.: Calculus of Variations Translator. In: Silverman, R.A. (ed.). Prentice Hall, Englewood Cliffs (1963)Google Scholar
  12. 12.
    Gubbiotti, G., Nucci, M.C.: J. Nonlinear Math. Phys. 21, 248–264 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hydon, P.E.: Symmetry Methods for Differential Equations: A Beginnner’s Guide. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  14. 14.
    Lutzky, M.: J. Phys. A Math. Gen. 11, 249–258 (1978)CrossRefGoogle Scholar
  15. 15.
    Mohanasubha, R., Sabiya Shakila, M.I., Senthilvelan, M.: Commun. Nonlinear. Sci. 19, 799–806 (2014)Google Scholar
  16. 16.
    Muriel, C., Romero, J.L.: \(\lambda \)-symmetries and linearization of ordinary differential equations through nonlocal transformations. PreprintGoogle Scholar
  17. 17.
    Muriel, C., Romero, J.L.: IMA J. Appl. Math. 66, 111 (2001)Google Scholar
  18. 18.
    Muriel, C., Romero, J.L.: Theor. Math. Phys. 133, 1565 (2002)CrossRefGoogle Scholar
  19. 19.
    Muriel, C., Romero, J.L.: J. Phys. A Math. Theor. 42, 365207 (2009)CrossRefGoogle Scholar
  20. 20.
    Muriel, C., Romero, J.L.: SIGMA 8, 106 (2012)Google Scholar
  21. 21.
    Noether, E.: Gttingen Nachrichten Mathematik-physik Klasse 2, 235 (1918)Google Scholar
  22. 22.
    Nucci, M.C., Tamizhmani, K.M.: J. Nonlinear Math. Phys. 17, 167 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)CrossRefGoogle Scholar
  24. 24.
    Pandey, S.N., Bindu, P.S., Senthilvelan, M., Lakshmanan, M.: J. Math. Phys. 50, 102701 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Polat, G.G., Özer, T.: Nonlinear Dyn. 85, 1571–1595 (2016)CrossRefGoogle Scholar
  26. 26.
    Pucci, E., Saccomandi, G.: J. Phys. A Math. Gen. 35, 6145 (2012)CrossRefGoogle Scholar
  27. 27.
    Yasar, E.: Mathematical Problems in Engineering, Article ID 916437, 10 pp. (2011)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • R. Mohanasubha
    • 1
  • V. K. Chandrasekar
    • 2
  • M. Senthilvelan
    • 3
  • M. Lakshmanan
    • 3
    Email author
  1. 1.Department of PhysicsAnna UniversityChennaiIndia
  2. 2.School of Electrical and Electronics EngineeringCentre for Nonlinear Science and Engineering, SASTRA UniversityThanjavurIndia
  3. 3.Centre for Nonlinear Dynamics, Bharathidasan UniversityTiruchirappalliIndia

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