Advertisement

Pramana

, Volume 48, Issue 1, pp 303–323 | Cite as

Control of spatiotemporal chaos: A study with an autocatalytic reaction-diffusion system

  • Nita Parekh
  • V Ravi Kumar
  • B D Kulkarni
Spatio-Temporal Chaos, Synchronization And Control

Abstract

The characterization of chaotic spatiotemporal dynamics has been studied for a representative nonlinear autocatalytic reaction mechanism coupled with diffusion. This has been carried out by an analysis of the Lyapunov spectrum in spatiallylocalised regions. The linear scaling relationships observed in the invariant measures as a function of thesub-system size have been utilized to assess the controllability, stability and synchronization properties of the chaotic dynamics. The dynamical synchronization properties of this high-dimensional system has been analyzed using suitable Lyapunov functionals. The possibility of controlling spatiotemporal chaos for relevant objectives using available noisy scalar time-series data with simultaneous self-adaptation of the control parameter(s) has also been discussed.

Keywords

Spatiotemporal chaos reaction-diffusion systems autocatalysis Lyapunov exponents synchronization and control parametric self-adaptation 

PACS Nos

05.70 82.20 47.54 05.45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M C Cross and P C Hohenberg,Rev. Mod. Phys. 65, 851 (1993)CrossRefADSGoogle Scholar
  2. [2]
    Y C Lai and R L Winslow,Physica D74, 353 (1994)ADSGoogle Scholar
  3. [3]
    T Shinbrot,Nonlinear Science Today 3, 1 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    H Gang and H Kaifen,Phys. Rev. Lett. 71, 3794 (1993)CrossRefADSGoogle Scholar
  5. [5]
    E Kostelich, C Grebogi, E Ott and J A Yorke,Phys. Rev. E47, 305 (1993)ADSMathSciNetGoogle Scholar
  6. [6]
    D Auerbach,Phys. Rev. Lett. 72, 1184 (1994)CrossRefADSGoogle Scholar
  7. [7]
    L Poon and C Grebogi,Phys. Rev. Lett. 75, 4023 (1995)CrossRefADSGoogle Scholar
  8. [8]
    A Karma,Phys. Rev. Lett. 71, 1103 (1993)zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. [9]
    M Bär and M Eiswirth,Phys. Rev. E48, R1635 (1993)Google Scholar
  10. [10]
    R Imbihl and G Ertl,Chem. Rev. 95, 697 (1995)CrossRefGoogle Scholar
  11. [11]
    B I Shraimanet al, Physica D57, 241 (1992)ADSMathSciNetGoogle Scholar
  12. [12]
    I Aranson, L Aranson, L Kramer and A Weber,Phys. Rev. A46, 2992 (1992)ADSGoogle Scholar
  13. [13]
    I Aranson, H Levine and L Tsimring,Phys. Rev. Lett. 72, 2561 (1994)CrossRefADSGoogle Scholar
  14. [14]
    J F Lindner, B K Meadows, W L Ditto, M E Inchiosa and A R Bulsara,Phys. Rev. Lett. 75, 3 (1995)CrossRefADSGoogle Scholar
  15. [15]
    Y Braiman, J F Lindner and W L Ditto,Nature (London) 378, 465 (1996)CrossRefADSGoogle Scholar
  16. [16]
    H D I Abarbanel, R Brown, J J Sidorowich and L Tsimring,Rev. Mod. Phys. 65, 1331 (1993)CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    H Fujisaka and T Yamada,Prog. Theor. Phys. 69, 32 (1983)zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. [18]
    L M Pecora and T L Carroll,Phys. Rev. Lett. 64, 821 (1990)CrossRefADSMathSciNetGoogle Scholar
  19. [19]
    L M Pecora and T L Carroll,Phys. Rev. A44, 2374 (1991)ADSGoogle Scholar
  20. [20]
    R He and P G Vaidya,Phys. Rev. A46, 7387 (1992)ADSMathSciNetGoogle Scholar
  21. [21]
    K M Cuomo and A V Oppenheim,Phys. Rev. Lett. 71, 65 (1993)CrossRefADSGoogle Scholar
  22. [22]
    M Ding and E Ott,Phys. Rev. E49, R945 (1994)Google Scholar
  23. [23]
    C W Wu and L O Chua,Int. J. Bifur. Chaos 4, 979 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    J F Heagy, T L Carroll and L M Pecora,Phys. Rev. E50, 1874 (1994)ADSGoogle Scholar
  25. [25]
    N F Rulkov, M M Sushchik, L S Tsimring and H D I Abarbanel,Phys. Rev. E51, 980 (1995)ADSGoogle Scholar
  26. [26]
    L Kocarev and U Parlitz,Phys. Rev. Lett. 76, (1996)Google Scholar
  27. [27]
    T C Newell, P M Alsing, A Gavrielides and V Kovanis,Phys. Rev. Lett. 72, 1647 (1994)CrossRefADSGoogle Scholar
  28. [28]
    T C Newell, P M Alsing, A Gavrielides and V Kovanis,Phys. Rev. E49, 313 (1994)ADSGoogle Scholar
  29. [29]
    E Ott, C Grebogi and J A Yorke,Phys. Rev. Lett. 64, 1196 (1990)zbMATHCrossRefADSMathSciNetGoogle Scholar
  30. [30]
    Y C Lai and C Grebogi,Phys. Rev. E47, 2357 (1993)ADSMathSciNetGoogle Scholar
  31. [31]
    P Grassberger,Phys. Scr. 40, 346 (1989)CrossRefADSGoogle Scholar
  32. [32]
    G Mayer-Kress and K Kaneko,J. Stat. Phys. 54, 1489 (1989)CrossRefADSMathSciNetGoogle Scholar
  33. [33]
    A Torcini, A Politi, G Puccioni and G Alessandro,Physica 53, 85 (1991)zbMATHMathSciNetGoogle Scholar
  34. [34]
    H Chate, G Grinstein and L H Tang,Phys. Rev. Lett. 74, 912 (1995)CrossRefADSGoogle Scholar
  35. [35]
    M Bauer, H Heng and W Martienssen,Phys. Rev. Lett. 71, 521 (1993)CrossRefADSGoogle Scholar
  36. [36]
    N Parekh, V Ravi Kumar and B D Kulkarni, (submitted)Google Scholar
  37. [37]
    Y C Lai and C Grebogi,Phys. Rev. E52, 1894 (1994)ADSMathSciNetGoogle Scholar
  38. [38]
    J Warncke, M Bauer and W Martienssen,Europhys. Lett. 25, 323 (1994)CrossRefADSGoogle Scholar
  39. [39]
    J H Peng, E J Ding, M Ding and W Yang,Phys. Rev. Lett. 76, 904 (1996)CrossRefADSGoogle Scholar
  40. [40]
    N Parekh, V Ravi Kumar and B D Kulkarni,Physica A224, 369 (1996)ADSGoogle Scholar
  41. [41]
    V Castets, E Dulos, J Boissonade and P De Kepper,Phys. Rev. Lett. 64, 2953 (1990)CrossRefADSGoogle Scholar
  42. [42]
    Y Kuramoto,Chemical oscillations, waves and turbulence (Berlin: Springer) (1984)zbMATHGoogle Scholar
  43. [43]
    G Nicolis,J. Phys. C2, SA47 (1990)Google Scholar
  44. [44]
    W Ouyang and H L Swinney,Nature (London) 352, 610 (1991)CrossRefADSGoogle Scholar
  45. [45]
    P Gray and S Scott,Chem. Eng. Sci. 38, 29 (1983)CrossRefGoogle Scholar
  46. [46]
    J E Pearson,Science 261, 189 (1993)CrossRefADSGoogle Scholar
  47. [47]
    K J Lee, W D McCormick, J E Pearson and H L Swinney,Nature (London) 369, 215 (1994)CrossRefADSGoogle Scholar
  48. [48]
    N Parekh, V Ravi Kumar and B D Kulkarni,Phys. Rev. E52, 5100 (1995)ADSGoogle Scholar
  49. [49]
    D T Lynch,Chem. Engg. Sci. 47, 4435 (1992)CrossRefGoogle Scholar
  50. [50]
    D Horváth, Valery Petrov, S K Scott and K Showalter,J. Chem. Phys. 98, 6332 (1993)CrossRefADSGoogle Scholar
  51. [51]
    V Petrov, S K Scott and K Showalter,Philos. Trans. R. Soc. London A347, 631 (1994)ADSGoogle Scholar
  52. [52]
    J Argyris, G Faust and M Haase,An exploration of chaos (Elsevier Science B V, Amsterdam, 1994)zbMATHGoogle Scholar
  53. [53]
    J L Kaplan and J A Yorke,Lecture notes in mathematics 730, 204 (1979)MathSciNetCrossRefGoogle Scholar
  54. [54]
    S N Rasband,Chaotic dynamics of nonlinear systems (Wiley-Interscience, 1989)Google Scholar
  55. [55]
    Y B Pesin,Russ. Math. Sur. 32, 55 (1977)CrossRefMathSciNetGoogle Scholar
  56. [56]
    H Tong,Nonlinear time series: A dynamical system approach (Clarendon Press, Oxford, 1990)Google Scholar
  57. [57]
    B A Huberman and E Lumer,IEEE Trans. Circuits Syst. 37, 547 (1990)CrossRefGoogle Scholar
  58. [58]
    S Sinha and R Ramaswamy,Physica D43, 118 (1990)ADSMathSciNetGoogle Scholar
  59. [59]
    V Ravi Kumar, B D Kulkarni and P B Deshpande,Proc. R. Soc. London Ser. A433, 711 (1991)ADSGoogle Scholar
  60. [60]
    S Rajashekar and M Lakshmanan,Int. J. Bifur. Chaos 2, 201 (1992)CrossRefGoogle Scholar
  61. [61]
    K Pyragas,Phys. Lett. A181, 203 (1993)ADSGoogle Scholar
  62. [62]
    J K Bandyopadhyay, V Ravi Kumar, B D Kulkarni and P Bhattacharya,Chem. Engg. Sci. 48, 3545 (1993)CrossRefGoogle Scholar
  63. [63]
    H K Qammer, F Mossayebi and L Murphy,Phys. Lett. A178, 279 (1993)ADSGoogle Scholar
  64. [64]
    D Vassiliadis,Physica D71, 319 (1994)ADSGoogle Scholar
  65. [65]
    J K John and R E Amritkar,Phys. Rev. E49, 4843 (1994)ADSGoogle Scholar
  66. [66]
    U Pralitz,Phys. Rev. Lett. 76, 1232 (1996)CrossRefADSGoogle Scholar
  67. [67]
    H G Bock,Progress in scientific computing (Birkhäuser, Boston)2, 95 (1983)Google Scholar
  68. [68]
    E Baake, M Baake, H G Bock and K M Briggs,Phys. Rev. A45, 5524 (1992)ADSGoogle Scholar
  69. [69]
    J C Principe, A Rathie and J M Huo,Int. J. Bifur. Chaos 2, 989 (1992)zbMATHCrossRefGoogle Scholar
  70. [70]
    S A Billings and S Chen,Neural networks and system identification (Peter Peregrinus, London, 1992) p. 181Google Scholar

Copyright information

© Indian Academy of Sciences 1997

Authors and Affiliations

  • Nita Parekh
    • 1
  • V Ravi Kumar
    • 1
  • B D Kulkarni
    • 1
  1. 1.Chemical Engineering DivisionNational Chemical LaboratoryPuneIndia

Personalised recommendations