Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Parametric disorder effects on a subcritical stationary bifurcation under nonlinear gradient term

  • 45 Accesses

Abstract

Effects of harmonic modulation of the threshold of the bifurcation are investigated in the one-dimensional cubic-quintic Ginzburg–Landau equation with real coefficients. We analyze the effects of the nonlinear gradient term which is of same order as the quintic term in the Ginzburg–Landau equation. Above the threshold, the nonlinear part of equation solutions are determined by the Poincaré–Lindstedt expansion approach. We show that for small values of the coefficient of the nonlinear gradient term, the stationary nonlinear solution change, the slope of the Nusselt number increases, while the curvature decreases with increasing values of the modulation amplitude.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    M.C. Cross, C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)

  2. 2.

    K. Nosaki, N. Bekki, Phys. Rev. Lett. 51, 2171 (1983)

  3. 3.

    I.S. Aranson, L. Kramer, Rev. Mod. Phys. 74, 99 (2002)

  4. 4.

    P. Kolonder, R.W. Walder, A. Passner, C.M. Surko, J. Fluid Mech. 163, 195 (1986)

  5. 5.

    N. Garnier, A. Chiffaudel, F. Daviaud, Phys. Rev. Lett. 88, 134501 (2002)

  6. 6.

    L. Nana, A.B. Ezersky, I. Mutabazi, Proc. R. Soc. A 465, 2251 (2009)

  7. 7.

    P. Bot, I. Mutabazi, Eur. Phys. J. B 13, 141 (2000)

  8. 8.

    I. Mutabazi, J.J. Hegseth, C.D. Andereck, J.E. Wesfreid, Phys. Rev. Lett. 64, 1729 (1990)

  9. 9.

    A. Joets et al., in Propagation in Systems far from Equilibrium (Springer, New York, 1988), Vol. 176

  10. 10.

    M. Ipsen, L. Kramer, P.G. Sorensen, Phys. Rep. 337, 193 (2000)

  11. 11.

    A.S. Mikhailov, K. Showalter, Phys. Rep. 425, 79 (2006)

  12. 12.

    L.M. Hocking , K. Stewartson, Proc. R. Soc. Lond. A 326, 289 (1972)

  13. 13.

    N.R. Pereira, L. Stenflo, Phys. Fluids 20, 1733 (1977)

  14. 14.

    K. Nosaki, N. Bekki, J. Phys. Soc. Jpn. 53, 1581 (1984)

  15. 15.

    N. Bekki, K. Nosaki, Phys. Lett. A 110, 133 (1985)

  16. 16.

    F. Cariello , M. Tabor, Physica D 39, 77 (1989)

  17. 17.

    R. Conte , M. Musette, Physica D 69, 1 (1993)

  18. 18.

    T. Kapitula, Nonlinearity 9, 669 (1996)

  19. 19.

    A.J. Bernoff, Physica D 30, 363 (1988)

  20. 20.

    B.P. Luce, Physica D 84, 553 (1995)

  21. 21.

    H. Chaté, Physica D 86, 238 (1995)

  22. 22.

    P. Kolodner, S. Slimani, N. Aubry, R. Lima, Physica D 85, 165 (1995)

  23. 23.

    L. R. Keefe, Stud. Appl. Math. 73, 91 (1985)

  24. 24.

    Y. Kuramoto, Chemical Oscillations Waves and Turbulence, (Springer, Berlin, 1984)

  25. 25.

    P. Coullet, L. Gil, J. Lega, Phys. Rev. Lett. 62, 1619 (1989)

  26. 26.

    S. Bottin, J. Lega, Eur. Phys J. B 5, 299 (1998)

  27. 27.

    M.W. Limi, T.C. Kofane, Int. J. Non-Linear Mech. 82, 75 (2016)

  28. 28.

    M. Hammele, S. Schuler, W. Zimmermann, Physica D 218, 139 (2006)

  29. 29.

    H. Herwig, Entropy 18, 1 (2016)

  30. 30.

    S.E. Jones, Int. J. Non-linear Mech. 13, 125 (1978)

  31. 31.

    T.D. Burton, Int. J. Non-linear Mech. 19, 397 (1984)

  32. 32.

    Y.K. Cheung, S.H. Cheun, S.L. Lau, Int. J. Non-linear Mech. 26, 367 (1991)

  33. 33.

    J.H. He, Int. J. Non-linear Mech. 37, 309 (2002)

  34. 34.

    J.H. He, Int. J. Non-linear Mech. 37, 315 (2002)

  35. 35.

    T.D. Burton, Z. Rahman, Int. J. Non-linear Mech. 21, 135 (1986)

  36. 36.

    V. Marincaa, N. Herisanu, J. Sound Vib. 329, 1450 (2010)

  37. 37.

    A. Belendez, Comput. Math. Appl. 58, 2267 (2009)

  38. 38.

    A.Y.T. Leung, Z.J. Guo, J. Sound Vib. 325, 287 (2009)

  39. 39.

    M. Sheikholeslami, H.R. Ashorynejad, D.D. Ganji, A. Yildirim, Sci. Iranica B. 19, 437 (2012)

  40. 40.

    M. Suleman, Q. Wu, Adv. Math. Phys. 5, 1 (2015)

  41. 41.

    A. Belendez, C. Pascual, T. Belendez, A. Hernandez, Nonlinear Anal. 10, 416 (2009)

  42. 42.

    A. Belendez, C. Pascual, M. Ortuno, T. Belendez, S. Gallego, Nonlinear Anal. 10, 601 (2009)

  43. 43.

    T. Ozis, A. Yildirim, J. Sound Vib. 301, 415 (2007)

  44. 44.

    J.H. He, Commun. Nonlinear Sci. Numer. Simul. 4, 81 (1999)

  45. 45.

    R. Azami, D.D. Ganji, H. Babazadeh, A.G. Dvavodi, S.S. Ganji, Int. J. Mod. Phys. B. 23, 5915 (2009)

  46. 46.

    E.Z. Alex, C.A. Rodrguez, O.M. Romero, Appl. Math. Comp. 218, 11112 (2012)

  47. 47.

    M. Sheikholeslami, M. Azimi, D.D. Ganji, Int. J. Comput. 16, 246 (2015)

  48. 48.

    M.R. Akbari, D.D. Ganji, A. Majidian, A.R. Ahmadi, Front Mech. Eng. 9, 177 (2014)

  49. 49.

    M. Daeichin, M.A. Ahmadpoor, H. Askari, A. Yildirim, Asian Eur. J. Math. 6, 1350019 (2013)

  50. 50.

    D. Younesian, H. Askari, Z. Saadatnia, M. Kalami Yazdi, Comput. Math. Appl. 59, 3222 (2010)

  51. 51.

    L. Zhao, Comput. Math. Appl. 58, 2477 (2009)

  52. 52.

    X.C. Cai, W.Y. Wu, Comput. Math. Appl. 58, 2358 (2009)

  53. 53.

    B.M. Ikramul, M.S. Alam, M.M. Rahman, J. Egypt Math. Soc. 21, 142 (2013)

  54. 54.

    J.H. He, Chaos Solitons Fract. 34, 1430 (2007)

  55. 55.

    A. Yildirim, H. Askari, M.K. Yazdi, Y. Khan, Appl. Math. Lett. 25, 1729 (2012)

  56. 56.

    M.P. Solon, J.P.H. Esguerra, Phys. Lett. A 372, 6608 (2008)

  57. 57.

    R.E. Mickens, Oscillation in Planar Dynamic Systems (World Scientific, Singapore, 1996)

  58. 58.

    R.E. Mickens, J. Sound Vib. 111, 515 (1986)

  59. 59.

    S.B. Yamgoue, J.R Bogning, A.K. Jiotsa, T.C. Kofane, Phys. Scr. 81, 035003 (2010)

  60. 60.

    S.B. Yamgoue, T.C. Kofane, Int. J. Non-linear Mech. 41, 1248 (2006)

  61. 61.

    L. Chen, N. Goldenfeld, Y. Oono, Phys. Rev. E 54, 376 (1996)

  62. 62.

    A. Sarkar, J.K. Bhattacharjee, S. Chakraborty, D.B. Banerjee, Eur. Phys. J. D 64, 479 (2011)

  63. 63.

    N. Goldenfeld, O. Martin, Y. Oono, F. Liu, Phys. Rev. Lett. 64, 1361 (1990)

  64. 64.

    G.C. Paquette, L.Y. Chen, N. Goldenfeld, Y. Oono, Phys. Rev. Lett. 72, 76 (1994)

Download references

Author information

Correspondence to Martine Limi Wokwenmendam.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wokwenmendam, M.L., Kofane, T.C. Parametric disorder effects on a subcritical stationary bifurcation under nonlinear gradient term. Eur. Phys. J. B 91, 227 (2018). https://doi.org/10.1140/epjb/e2018-80699-2

Download citation

Keywords

  • Statistical and Nonlinear Physics