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Control and managing of localized states in two-dimensional systems with periodic forcing

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Abstract

We study the formation of localized structures in two-dimensional systems with periodic forcing, showing that these types of systems provide an adequate framework for the study and control of localized structures. Theoretically, we introduce a dissipative ϕ 4 model as a prototype for a bistable spatially forced system, and we show that with different spatial forcings of small amplitudes, such as square or hexagonal grids, this model exhibits a family of localized structures. By changing the forcing parameters, we control the bistability between the various induced patterns. Experimentally, based on an optical feedback with spatially amplitude-modulated beam, we set-up a two-dimensional forced experiment in a nematic liquid crystal cell. By changing the forcing parameters, the system exhibits a family of localized structures that are confirmed by numerical simulations for the average liquid crystal tilt angle.

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References

  1. See, e.g., L.M. Pismen, Patterns and Interfaces in Dissipative Dynamics (Springer Series in Synergetics, Berlin, Heidelberg, 2006), and references therein

  2. R.A. Fisher, Ann. Eugenics 7, 335 (1937)

    Google Scholar 

  3. A. Kolmogorov, I. Petrovsky, Piskunov, Bull. Univ. Moskou Ser. Int. Se. 7, 1 (1937)

    Google Scholar 

  4. J.D. Murray, Mathematical Biology (Springer-Verlag, Berlin, 1993)

  5. P. Kife, Mathematical Aspects of Reacting and Diffusing System, edited by S. Levin, Lecture Notes in Biomathematics 28 (Springer-Verlag, New York, 1979)

  6. D. Walgraef, Spatio-temporal pattern formation (Springer-Verlag, New York, 1997)

  7. G. Dee, J.S. Langer, Phys. Rev. Lett. 50, 383 (1983)

    Article  ADS  Google Scholar 

  8. P. Manneville, Dissipative structures and weak turbulence (Academic Press, San Diego, 1990)

  9. D.G. Aronson, H.F. Weinberger, Adv. Math. 30, 33 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  10. P. Collet, J.P. Eckmann, Instabilities and fronts in extended system (Princeton University Press, New Jersey, 1990)

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

    Article  ADS  Google Scholar 

  12. H.A. Eschenfelder, Magnetic Bubble Technology (Springer Verlag, Berlin, 1981)

  13. S. Pirkl, P. Ribière, P. Oswald, Liq. Cryst. 13, 413 (1993)

    Article  Google Scholar 

  14. Y.A Astrov, Y.A. Logvin, Phys. Rev. Lett. 79, 2983 (1997)

    Article  ADS  Google Scholar 

  15. K.-J. Lee, W.D. McCormick, J.E. Pearson, H.L. Swinney, Nature 369, 215 (1994)

    Article  ADS  Google Scholar 

  16. W.S. Edwards, S. Fauve, J. Fluid Mech. 278, 123 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  17. P.B. Umbanhowar, F. Melo, H.L. Swinney, Nature 382, 793 (1996)

    Article  ADS  Google Scholar 

  18. M.G. Clerc, P. Cordero, J. Dunstan, K. Huff, N. Mujica, D. Risso, G. Varas, Nat. Phys. 4, 249 (2008)

    Article  Google Scholar 

  19. R. Heinrichs, G. Ahlers, D.S. Cannell, Phys. Rev. A 35, 2761(R) (1987)

    Article  ADS  Google Scholar 

  20. P. Kolodner, D. Bensimon, C.M. Surko, Phys. Rev. Lett. 60, 1723 (1988)

    Article  ADS  Google Scholar 

  21. D.W. Mc Laughlin, J.V. Moloney, A.C. Newell, Phys. Rev. Lett. 51, 75 (1983)

    Article  ADS  Google Scholar 

  22. N.N. Rosanov, G.V. Khodova, Opt. Spectrosc. 65, 449 (1988)

    ADS  Google Scholar 

  23. T. Ackemann, W.J. Firth, G.L. Oppo, Adv. At. Mol. Opt. Phys. 57, 323 (2009)

    Article  Google Scholar 

  24. W. van Saarloos, P.C. Hohenberg, Phys. Rev. Lett. 64, 749 (1990)

    Article  ADS  Google Scholar 

  25. P.D. Woods, A.R. Champneys, Physica D 129, 147 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. G.W. Hunt, G.J. Lord, A.R. Champneys, Comput. Meth. Appl. Mech. Eng. 170, 239 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Y. Pomeau, Physica D 23, 3 (1986)

    Article  ADS  Google Scholar 

  28. P. Coullet, C. Riera, C. Tresser, Phys. Rev. Lett. 84, 3069 (2000)

    Article  ADS  Google Scholar 

  29. J. Burke, E. Knobloch, Phys. Rev. E 73, 056211 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  30. J. Burke, E. Knobloch, Chaos 17, 037102 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  31. M.G. Clerc, C. Falcon, Physica A 356, 48 (2005)

    Article  ADS  Google Scholar 

  32. U. Bortolozzo, M.G. Clerc, C. Falcon, S. Residori, R. Rojas, Phys. Rev. Lett. 96, 214501 (2006)

    Article  ADS  Google Scholar 

  33. W.J. Firth, L. Columbo, A.J. Scroggie, Phys. Rev. Lett. 99, 104503 (2007)

    Article  ADS  Google Scholar 

  34. U. Bortolozzo, M.G. Clerc, S. Residori, Phys. Rev. E 78, 036214 (2008)

    Article  ADS  Google Scholar 

  35. U. Bortolozzo, M.G. Clerc, S. Residori, New J. Phys. 11, 093037 (2009)

    Article  ADS  Google Scholar 

  36. M.G. Clerc, A. Petrossian, S. Residori, Phys. Rev. E 71, 015205(R) (2005)

    Article  ADS  Google Scholar 

  37. S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Addison-Wesley, Reading, Massachussets, 1994)

  38. R.E. Goldstein, G.H. Gunaratne, L. Gil, P. Coullet, Phys. Rev. A 43, 6700 (1991)

    Article  ADS  Google Scholar 

  39. P. Coullet, Int. J. Bifur. Chaos 12, 2445 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  40. F. Haudin, R.G. Elías, R.G. Rojas, U. Bortolozzo, M.G. Clerc, S. Residori, Phys. Rev. Lett. 103, 128003 (2009)

    Article  ADS  Google Scholar 

  41. U. Bortolozzo, M.G. Clerc, F. Haudin, R.G. Rojas, S. Residori, Adv. Nonlinear Opt. 2009, 926810 (2009)

    Google Scholar 

  42. D. Bensimon, B.I. Shraiman, V. Croquette, Phys. Rev. A 38, R5461 (1988)

    Article  ADS  Google Scholar 

  43. M.G. Clerc, D. Escaff, C. Falcon, E. Tirapegui, Eur. Phys. J. Special Topics 143, 171 (2007)

    Article  ADS  Google Scholar 

  44. T.J. Alexander, A.A. Sukhorukov, Y.S. Kivshar, Phys. Rev. Lett. 93, 063901 (2004)

    Article  ADS  Google Scholar 

  45. S. Residori, Phys. Rep. 416, 201 (2005)

    Article  ADS  Google Scholar 

  46. M.G. Clerc, S. Residori, C.S. Riera, Phys. Rev. E 63, 060701(R) (2001)

    Article  ADS  Google Scholar 

  47. U. Bortolozzo, S. Residori, Phys. Rev. Lett. 96, 037801 (2006)

    Article  ADS  Google Scholar 

  48. P.G. De Gennes, J. Prost, The Physics of Liquid Crystals, second edition (Oxford Science Publications, Clarendon Press, 1993)

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Authors and Affiliations

  1. Departmento de Física, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile

    M. G. Clerc

  2. INLN, Université de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles, 06560, Valbonne, France

    F. Haudin, S. Residori & U. Bortolozzo

  3. Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile

    R. G. Rojas

Authors
  1. M. G. Clerc
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  2. F. Haudin
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  3. S. Residori
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  4. U. Bortolozzo
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  5. R. G. Rojas
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Correspondence to M. G. Clerc.

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Clerc, M., Haudin, F., Residori, S. et al. Control and managing of localized states in two-dimensional systems with periodic forcing. Eur. Phys. J. D 59, 43–51 (2010). https://doi.org/10.1140/epjd/e2010-00084-9

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  • DOI: https://doi.org/10.1140/epjd/e2010-00084-9

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