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

  • Spatio-Temporal Chaos, Synchronization And Control
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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.

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References

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

    Article  ADS  Google Scholar 

  2. Y C Lai and R L Winslow,Physica D74, 353 (1994)

    ADS  Google Scholar 

  3. T Shinbrot,Nonlinear Science Today 3, 1 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. H Gang and H Kaifen,Phys. Rev. Lett. 71, 3794 (1993)

    Article  ADS  Google Scholar 

  5. E Kostelich, C Grebogi, E Ott and J A Yorke,Phys. Rev. E47, 305 (1993)

    ADS  MathSciNet  Google Scholar 

  6. D Auerbach,Phys. Rev. Lett. 72, 1184 (1994)

    Article  ADS  Google Scholar 

  7. L Poon and C Grebogi,Phys. Rev. Lett. 75, 4023 (1995)

    Article  ADS  Google Scholar 

  8. A Karma,Phys. Rev. Lett. 71, 1103 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. M Bär and M Eiswirth,Phys. Rev. E48, R1635 (1993)

  10. R Imbihl and G Ertl,Chem. Rev. 95, 697 (1995)

    Article  Google Scholar 

  11. B I Shraimanet al, Physica D57, 241 (1992)

    ADS  MathSciNet  Google Scholar 

  12. I Aranson, L Aranson, L Kramer and A Weber,Phys. Rev. A46, 2992 (1992)

    ADS  Google Scholar 

  13. I Aranson, H Levine and L Tsimring,Phys. Rev. Lett. 72, 2561 (1994)

    Article  ADS  Google Scholar 

  14. J F Lindner, B K Meadows, W L Ditto, M E Inchiosa and A R Bulsara,Phys. Rev. Lett. 75, 3 (1995)

    Article  ADS  Google Scholar 

  15. Y Braiman, J F Lindner and W L Ditto,Nature (London) 378, 465 (1996)

    Article  ADS  Google Scholar 

  16. H D I Abarbanel, R Brown, J J Sidorowich and L Tsimring,Rev. Mod. Phys. 65, 1331 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  17. H Fujisaka and T Yamada,Prog. Theor. Phys. 69, 32 (1983)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. L M Pecora and T L Carroll,Phys. Rev. Lett. 64, 821 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  19. L M Pecora and T L Carroll,Phys. Rev. A44, 2374 (1991)

    ADS  Google Scholar 

  20. R He and P G Vaidya,Phys. Rev. A46, 7387 (1992)

    ADS  MathSciNet  Google Scholar 

  21. K M Cuomo and A V Oppenheim,Phys. Rev. Lett. 71, 65 (1993)

    Article  ADS  Google Scholar 

  22. M Ding and E Ott,Phys. Rev. E49, R945 (1994)

  23. C W Wu and L O Chua,Int. J. Bifur. Chaos 4, 979 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. J F Heagy, T L Carroll and L M Pecora,Phys. Rev. E50, 1874 (1994)

    ADS  Google Scholar 

  25. N F Rulkov, M M Sushchik, L S Tsimring and H D I Abarbanel,Phys. Rev. E51, 980 (1995)

    ADS  Google Scholar 

  26. L Kocarev and U Parlitz,Phys. Rev. Lett. 76, (1996)

  27. T C Newell, P M Alsing, A Gavrielides and V Kovanis,Phys. Rev. Lett. 72, 1647 (1994)

    Article  ADS  Google Scholar 

  28. T C Newell, P M Alsing, A Gavrielides and V Kovanis,Phys. Rev. E49, 313 (1994)

    ADS  Google Scholar 

  29. E Ott, C Grebogi and J A Yorke,Phys. Rev. Lett. 64, 1196 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Y C Lai and C Grebogi,Phys. Rev. E47, 2357 (1993)

    ADS  MathSciNet  Google Scholar 

  31. P Grassberger,Phys. Scr. 40, 346 (1989)

    Article  ADS  Google Scholar 

  32. G Mayer-Kress and K Kaneko,J. Stat. Phys. 54, 1489 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  33. A Torcini, A Politi, G Puccioni and G Alessandro,Physica 53, 85 (1991)

    MATH  MathSciNet  Google Scholar 

  34. H Chate, G Grinstein and L H Tang,Phys. Rev. Lett. 74, 912 (1995)

    Article  ADS  Google Scholar 

  35. M Bauer, H Heng and W Martienssen,Phys. Rev. Lett. 71, 521 (1993)

    Article  ADS  Google Scholar 

  36. N Parekh, V Ravi Kumar and B D Kulkarni, (submitted)

  37. Y C Lai and C Grebogi,Phys. Rev. E52, 1894 (1994)

    ADS  MathSciNet  Google Scholar 

  38. J Warncke, M Bauer and W Martienssen,Europhys. Lett. 25, 323 (1994)

    Article  ADS  Google Scholar 

  39. J H Peng, E J Ding, M Ding and W Yang,Phys. Rev. Lett. 76, 904 (1996)

    Article  ADS  Google Scholar 

  40. N Parekh, V Ravi Kumar and B D Kulkarni,Physica A224, 369 (1996)

    ADS  Google Scholar 

  41. V Castets, E Dulos, J Boissonade and P De Kepper,Phys. Rev. Lett. 64, 2953 (1990)

    Article  ADS  Google Scholar 

  42. Y Kuramoto,Chemical oscillations, waves and turbulence (Berlin: Springer) (1984)

    MATH  Google Scholar 

  43. G Nicolis,J. Phys. C2, SA47 (1990)

  44. W Ouyang and H L Swinney,Nature (London) 352, 610 (1991)

    Article  ADS  Google Scholar 

  45. P Gray and S Scott,Chem. Eng. Sci. 38, 29 (1983)

    Article  Google Scholar 

  46. J E Pearson,Science 261, 189 (1993)

    Article  ADS  Google Scholar 

  47. K J Lee, W D McCormick, J E Pearson and H L Swinney,Nature (London) 369, 215 (1994)

    Article  ADS  Google Scholar 

  48. N Parekh, V Ravi Kumar and B D Kulkarni,Phys. Rev. E52, 5100 (1995)

    ADS  Google Scholar 

  49. D T Lynch,Chem. Engg. Sci. 47, 4435 (1992)

    Article  Google Scholar 

  50. D Horváth, Valery Petrov, S K Scott and K Showalter,J. Chem. Phys. 98, 6332 (1993)

    Article  ADS  Google Scholar 

  51. V Petrov, S K Scott and K Showalter,Philos. Trans. R. Soc. London A347, 631 (1994)

    ADS  Google Scholar 

  52. J Argyris, G Faust and M Haase,An exploration of chaos (Elsevier Science B V, Amsterdam, 1994)

    MATH  Google Scholar 

  53. J L Kaplan and J A Yorke,Lecture notes in mathematics 730, 204 (1979)

    Article  MathSciNet  Google Scholar 

  54. S N Rasband,Chaotic dynamics of nonlinear systems (Wiley-Interscience, 1989)

  55. Y B Pesin,Russ. Math. Sur. 32, 55 (1977)

    Article  MathSciNet  Google Scholar 

  56. H Tong,Nonlinear time series: A dynamical system approach (Clarendon Press, Oxford, 1990)

    Google Scholar 

  57. B A Huberman and E Lumer,IEEE Trans. Circuits Syst. 37, 547 (1990)

    Article  Google Scholar 

  58. S Sinha and R Ramaswamy,Physica D43, 118 (1990)

    ADS  MathSciNet  Google Scholar 

  59. V Ravi Kumar, B D Kulkarni and P B Deshpande,Proc. R. Soc. London Ser. A433, 711 (1991)

    ADS  Google Scholar 

  60. S Rajashekar and M Lakshmanan,Int. J. Bifur. Chaos 2, 201 (1992)

    Article  Google Scholar 

  61. K Pyragas,Phys. Lett. A181, 203 (1993)

    ADS  Google Scholar 

  62. J K Bandyopadhyay, V Ravi Kumar, B D Kulkarni and P Bhattacharya,Chem. Engg. Sci. 48, 3545 (1993)

    Article  Google Scholar 

  63. H K Qammer, F Mossayebi and L Murphy,Phys. Lett. A178, 279 (1993)

    ADS  Google Scholar 

  64. D Vassiliadis,Physica D71, 319 (1994)

    ADS  Google Scholar 

  65. J K John and R E Amritkar,Phys. Rev. E49, 4843 (1994)

    ADS  Google Scholar 

  66. U Pralitz,Phys. Rev. Lett. 76, 1232 (1996)

    Article  ADS  Google Scholar 

  67. H G Bock,Progress in scientific computing (Birkhäuser, Boston)2, 95 (1983)

    Google Scholar 

  68. E Baake, M Baake, H G Bock and K M Briggs,Phys. Rev. A45, 5524 (1992)

    ADS  Google Scholar 

  69. J C Principe, A Rathie and J M Huo,Int. J. Bifur. Chaos 2, 989 (1992)

    Article  MATH  Google Scholar 

  70. S A Billings and S Chen,Neural networks and system identification (Peter Peregrinus, London, 1992) p. 181

    Google Scholar 

Download references

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Authors and Affiliations

  1. Chemical Engineering Division, National Chemical Laboratory, 411 008, Pune, India

    Nita Parekh, V Ravi Kumar & B D Kulkarni

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  1. Nita Parekh
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  2. V Ravi Kumar
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  3. B D Kulkarni
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Parekh, N., Kumar, V.R. & Kulkarni, B.D. Control of spatiotemporal chaos: A study with an autocatalytic reaction-diffusion system. Pramana - J Phys 48, 303–323 (1997). https://doi.org/10.1007/BF02845637

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