Equivalent linearization finds nonzero frequency corrections beyond first order

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Abstract

We show that the equivalent linearization technique, when used properly, enables us to calculate frequency corrections of weakly nonlinear oscillators beyond the first order in nonlinearity. We illustrate the method by applying it to the conservative anharmonic oscillators and the nonconservative van der Pol oscillator that are respectively paradigmatic systems for modeling center-type oscillatory states and limit cycle type oscillatory states. The choice of these systems is also prompted by the fact that first order frequency corrections may vanish for both these types of oscillators, thereby rendering the calculation of the higher order corrections rather important. The method presented herein is very general in nature and, hence, in principle applicable to any arbitrary periodic oscillator.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    A.H. Nayfeh, Perturbation Methods (Wiley-VCH Verlag GmbH, Berlin, 2000)Google Scholar
  2. 2.
    D.-O. Jeon, K.R. Huang, J.-H. Jang, H. Jin, H. Jang, Phys. Rev. Lett. 114, 184802 (2015)ADSCrossRefGoogle Scholar
  3. 3.
    A. Chenciner, Scholarpedia 2, 2111 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    J.K. Bhattacharjee, D.S. Ray, Am. J. Phys. 84, 434 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    L.-Y. Chen, N. Goldenfeld, Y. Oono, Phys. Rev. E 54, 376 (1996)ADSCrossRefGoogle Scholar
  6. 6.
    N. Minorsky, in Introduction to Nonlinear Mechanics: Topological Methods, Analytical Methods, Nonlinear Resonance, Relaxation Oscillations, edited by J.W. Edwards (Ann Arbor, Michigan, 1947)Google Scholar
  7. 7.
    N.W. McLachlan, Ordinary Nonlinear Differential Equations in Engineering and Physical Sciences (Clarendon Press, Oxford, 1950)Google Scholar
  8. 8.
    T.K. Caughey, J. Acoust. Soc. Am. 35, 1706 (1963)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    G.D. Mahan, Many-Particle Physics (Physics of Solids and Liquids) (Springer, New York, 2000)Google Scholar
  10. 10.
    R. Sain, J.K. Bhattacharjee, Eur. J. Phys. 36, 055025 (2015)CrossRefGoogle Scholar
  11. 11.
    K. Banerjee, J.K. Bhattacharjee, H.S. Mani, Phys. Rev. A 30, 1118 (1984)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    D.W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations: Problems and Solutions: A Sourcebook for Scientists and Engineers (Oxford Texts in Applied and Engineering Mathematics) (Oxford University Press, Oxford, 2007)Google Scholar
  13. 13.
    J.J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems (Wiley-Interscience, New York, 1950)Google Scholar
  14. 14.
    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products (Corrected and Enlarged Edition) (Academic Press, 1980)Google Scholar
  15. 15.
    T. Kanamaru, Scholarpedia 2, 2202 (2007)ADSCrossRefGoogle Scholar
  16. 16.
    T. Shah, R. Chattopadhyay, K. Vaidya, S. Chakraborty, Phys. Rev. E 92, 062927 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    T. Shah, R. Chattopadhyay, S. Chakraborty, arXiv:1610.05218 (2016)
  18. 18.
    J.K. Bhattacharjee, A.K. Mallik, S. Chakraborty, Indian J. Phys. 81, 1115 (2007)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of PhysicsIndian Institute of Technology KanpurUttar PradeshIndia

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