ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Krylov Bogoliubov Type Analysis of Variants of the Mathieu Equation

Journal of Applied Nonlinear Dynamics 6(1) (2017) 57--77 | DOI:10.5890/JAND.2017.03.005

B. Shayak$^{1}$,$^{3}$; Pranav Vyas$^{2}$

$^{1}$ Department of Physics, Indian Institute of Technology Kanpur, NH-91, Kalyanpur Kanpur - 208016 Uttar Pradesh, INDIA

$^{2}$ Department of Mechanical Engineering, Indian Institute of Technology Kanpur, NH-91, Kalyanpur Kanpur–208016 Uttar Pradesh, INDIA

$^{3}$ Department of Theoretical and Applied Mechanics and Mechanical Engineering, Cornell University, Ithaca, New York –14853 US

Abstract

In this work we show that a Krylov-Bogoliubov type analysis is a powerful method for analysing variants of the Mathieu equation. We first demonstrate the technique by rederiving the results obtained by prior authors using different techniques and then apply it to a case where the system has a quasiperiodic drive (inhomogeneity) in addition to a quasiperiodic parametric term. A realistic system where such a forcing is present is an induction motor, so we adopt that as our model system to show the details of the method.

Acknowledgments

We are grateful to Professor RICHARD RAND for helpful discussion and suggestions which have greatly improved the quality of this manuscript. Shayak is also grateful to Kishore Vaigyanik Protsahan Yojana (KVPY) Government of India for a generous Fellowship.

References

1.  [1] Jordan, D.W. and Smith, P. (2007), Nonlinear Ordinary Differential Equations, 4th Edition, Oxford University Press, Oxford, UK.
2.  [2] Rand, R.H., Zounes, R. S., and Hastings, R. (1997), A Quasiperiodic Mathieu equation, The Richard Rand 50th Anniversary Volume of Nonlinear Dynamics, World Scientific, Singapure.
3.  [3] Zounes, R.S. (1997), An Analysis of the nonlinear quasiperiodic Mathieu equation, Doctoral Dissertation, Cornell University.
4.  [4] Zounes, R.S. and Rand, R.H. (1998), Transition curves for the quasiperiodic Mathieu equation, SIAM Journal of Applied Mathematics, 58, 1094.
5.  [5] Rand, R.H., Guennoun, K., and Belhaq, M. (2003), 2:2:1 Resonance in the quasiperiodic Mathieu equation, Nonlinear Dynamics 31, 367.
6.  [6] Abouhazim, N., Rand, R.H., and Belhaq, M. (2006), The Damped nonlinear quasiperiodic Mathieu equation near 2:2:1 resonance, Nonlinear Dynamics 45, 237.
7.  [7] Waters, T.J. (2010), Stability of a two-dimensional Mathieu-type system with quasiperiodic coefficients, Nonlinear Dynamics 60, 341.
8.  [8] Kotowski, G. (1943), Losungen der inhomogenen Mathieuschen Differential-gleichung mit periodischer storfunktion beliebiger frequenz (mit besonderer berucksichtigung der resonanzlosungen), Zeitschrift fur Angewande Mathematik und Mechanik 23, 213.
9.  [9] Belhaq, M. and Houssni, M. (1999), Quasiperiodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations, Nonlinear Dynamics 18, 1.
10.  [10] Belhaq, M., Kirrou, I., and Mokni, M. (2013), Periodic and quasiperiodic galloping of a wind excited tower under external excitation Nonlinear Dynamics 74, 849.
11.  [11] Rand, R.H. (1984), Computer Algebra in Applied Mathematics : an Introduction to MACSYMA, Boston, Massachusetts, USA.
12.  [12] Kovacs, P.K. and Racz, I. (1959), Transient Behaviour in Electric Machinery, Verlag der Ungarische Akademie der Wissenschaften.
13.  [13] Krishnan, R. (2010), Electric Motor Drives - Modeling, Analysis and Control, PHI Learning Private Limited, New Delhi.
14.  [14] Cubero, D., Pascual, J.C., and Renzoni, F.(2014), Irrationality and quasiperiodicity in driven nonlinear systems, Physical Review Letters 112, 174102.
15.  [15] Nayfeh, A.H. (2011), Introduction to Perturbation Techniques, Wiley Interscience, New Jersey, USA.
16.  [16] Perko, L.M. (1968), Higher order averaging and related methods for perturbed periodic and quasiperiodic systems, SIAM Journal of Applied Mathematics 17, 698.
17.  [17] Yu, P., Desai, Y.M., Popplewell, N., and Shah, A.H. (1996), The Krylov, Bogoliubov and Galerkin methods for nonlinear oscillations, Journal of Sound and Vibration 192, 413.
18.  [18] Ruelle, D. and Takens, F. (1971), On the Nature of turbulence, Communications in Mathematical Physics 20, 167.
19.  [19] Stavans, J., Heslot, F., and Libchaber, A. (1985), Fixed winding number and the quasiperiodic route to chaos in a convective fluid, Physical Review Letters 55, 596.