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Control of Flow Instabilities and Unsteady Flows pp 43–117Cite as

Optimal Control and Other Complex Systems Paradigms in the Context of Turbulent Flows

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Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 369))

Abstract

Complex systems are high dimensional, nonlinear systems far from equilibrium. Turbulent fluids can therefore be considered as a typical example for complex systems. In these lectures, we present new principles and methodologies of complex systems research, that may help to understand and control the dynamics of turbulent flows. The dynamics of a high dimensional complex system can in many cases be estimated from simple models such as coupled maps dynamics, low dimensional systems of ODEs, singular motions, or genetic algorithms. The reduction of dimensionality means that a few order parameters govern the dynamics of the complex system.

While the mentioned models are simple, they still may contain a large number of parameters. These parameters can be determined from experiments if the experimental dynamics is complex, e.g. aperiodic or chaotic. However, if the experimental dynamics is simple, e.g. a fixed point dynamics, it is necessary to perturb it in order to reconstruct the model. Of course, perturbations are also necessary to control a system. The problem with these perturbations is that they may activate additional variables which are not included in the simple model. Thus, the simple model is in general no longer valid for the perturbed system. From the principle of the dynamical key, we derive a special class of aperiodic forcing functions, that overcome this problem and make it possible to investigate a complex system with spectroscopic methods based on the simple model. We study different methods of system identification and control of complex systems.

Finally, we assign a physical meaning to some of the attractors of the simple models. We formulate the physical meanings in terms of variation principles, including the principle of minimum resistance, the leadership and the simplification paradigm.

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References

  1. H. Haken, Synergetics, an introduction (Springer, Berlin, 1983), chapt. 7.

    Google Scholar 

  2. R.P. Lippmann, IEEE ASSP Mag., p. 4, April (1987).

    Google Scholar 

  3. L.A. Zadeh, Fuzzy sets and applications ( Wiley, New York, 1987 ).

    MATH  Google Scholar 

  4. J.H. Holland, Adaptation in natural and neural systems ( Univ. of Michigan Press, Ann Arbor, 1975 ).

    Google Scholar 

  5. C.G. Langton in Artificial life, edited by C.G. Langton. ( Addison-Wesley, Redwood City, CA 1989 ).

    Google Scholar 

  6. H.G. Schuster, Deterministic Chaos ( Physik-Verlag, Weinheim, 1984 ).

    MATH  Google Scholar 

  7. B.B. Mandelbrot, The fractal geometry of nature ( Freeman, San Francisco, 1983 ).

    Google Scholar 

  8. D.K. Campbell, A.C. Newell, R.J. Schrieffer, H. Segur, Physica D 18, 1 (1986); R.K. Bullough and J.P. Caudrey Eds., Solitons ( Springer, New York, 1980 ).

    Google Scholar 

  9. R. Thom, Structural stability and morphogenesis (W.A. Benjamin, Reading, Mass. 1975 ).

    Google Scholar 

  10. P.C. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A 38, 364 (1988).

    MATH  MathSciNet  Google Scholar 

  11. E.A. Jackson Perspectives of nonlinear dynamics (Cambridge University Press, 1991)

    Google Scholar 

  12. M.J. Feigenbaum, J. Stat. Phys 19, 25 (1978).

    MATH  MathSciNet  Google Scholar 

  13. A: Hübler, E. Löscher, Naturwissenschafte n 76, 67 (1989); A. Hübler, Helv. Phys. Acta 62, 543 (1989).

    Google Scholar 

  14. L.R. Keefe, Phys. Fluids A 5, 931 (1993).

    MathSciNet  Google Scholar 

  15. F.Ohle,H. Eckelmann, IUTAM-Symposium on Bluff Body Wakes, Dynamics, and Instabilities (Springer,Heidelberg, 1993), p.241.

    Google Scholar 

  16. B.Rueckerl,W. Eberl, A. Hübler, and E. Lüscher, Helv. Phys. Acta 62, 290 (1988).

    Google Scholar 

  17. F. Ohle, M. Lange, Phys. Rev. E 53, 531 (1996).

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  19. E. Berger,Phys. Fluids 10, 191 (1967).

    Google Scholar 

  20. P. Huerre and P.A. Monkewitz, Annu. Rev. Fluid. Mech. 22, 473 (1990).

    MathSciNet  Google Scholar 

  21. L.D. Landau C. R. Dokl. Acad. Sci. USSr 44, 311 (1944).

    MATH  Google Scholar 

  22. D. Ruelle and F. Takens, Commun. Math. Phys. 20, 167 (1971).

    MATH  MathSciNet  Google Scholar 

  23. S. Grossmann, Phys. Blätter 7 (1995).

    Google Scholar 

  24. B.V. Chirikov, Phys. Rep. 52, 263 (1979).

    MathSciNet  Google Scholar 

  25. C. Wargitsch, A. Hübler, Phys. Rev. E 51, 1508 (1995).

    MathSciNet  Google Scholar 

  26. K. Chang, A. Kodogergiou, A. Hübler, E.A. Jackson, Physica D 51, 77 (1991)

    Google Scholar 

  27. A. Hübler, U. Stoffregen, R. Wittmann, T. Nagata

    Google Scholar 

  28. S. Krempl, T. Eisenhammer, G. Mayer-Kress, P.W. Milonni, A. Huebler, Phys. Rev. Lett. 69, 430 (1992).

    Google Scholar 

  29. F. Yamaguchi, K. Kawamura, A. Hübler, Jpn. J. Appl. Phys. 34, L105 (1995).

    Google Scholar 

  30. B. Merte, G. Hadwich, B. Binias, P. Deisz, A. Hübler, E. Löscher, Helv. Phys. Acta 62, 294 (1989); M. Athelogou, P. Deisz, B. Merte, A. Hübler, E. Löscher, Heiv. Phys. Acta 62, 250 ( 1989 ); M. Dueweke, B. Merte, A.Hübler, preprint.

    Google Scholar 

  31. M. Dueweke, U. Dierker, A.Hübler, to appear in Phys. Rev. E (1996).

    Google Scholar 

  32. A. Hübler, D. Pines, in Complexity: From Metapher to Reality edited by G. Cowan, D. Pines, G. Meltzer (Addison-Wesley, 1993 ).

    Google Scholar 

  33. D. Pierre, A.W. Hübler, Physica D 75, 343 (1994).

    MATH  Google Scholar 

  34. A. Hübler in Modeling Complex Phenomena edited by L. Lam, A. Naroditsky ( Springer, New York, 1992 ), P. 5.

    Google Scholar 

  35. H. Haken, Advanced Synergetics ( Springer, New York, 1983 ).

    MATH  Google Scholar 

  36. V.I. Arnold, Mathematical methods of classical mechanics, 4th ed. ( Springer, New York, 1984 ), p. 4.

    Google Scholar 

  37. W. Stelzel, T. Kautzky, A. Hübler, E. Löscher, Heiv. Phys. Acta 61, 224 (1988); R. Shermer, F. Dinkelacker, A.

    Google Scholar 

  38. L. D. Landau, E. M. Lifshitz Lehrbuch der Theoretischen Physik I, Mechanik ( Akademie-Verlag, Berlin, 1979 ).

    Google Scholar 

  39. J.P. Crutchfield, K. Young, Phys. Rev. Lett. 63, 105 (1989).

    MathSciNet  Google Scholar 

  40. B. Plapp, A. Hübler, Phys. Rev. Lett. 65, 2302 (1990).

    Google Scholar 

  41. J.Levy, M.S. Sherwin, Phys. Rev. Lett. 70, 2597 (1993).

    Google Scholar 

  42. B.R. Noack, F. Ohle, H. Eckelmann, Physica D 56, 389 (1992).

    MATH  MathSciNet  Google Scholar 

  43. A.J. Lichtenberg and M.A. Lieberman, Regular and stoch astic motion ( Springer, New York, 1982 ).

    Google Scholar 

  44. A.H. Nayfeh and D.T. Mook, Nonlinear oscillations ( John Wiley and Sons, New York, 1976 ).

    Google Scholar 

  45. L.E. Arsnault, A.W. Hübler, Phys. Rev. E 51, 3561 (1995).

    MathSciNet  Google Scholar 

  46. L.E. Arsenault Coupled Oscillators near Resonance, PhD thesis at University of Illinois at Urbana-Champaign, 1996.

    Google Scholar 

  47. J.M. Greene, J. Math. Phys. 20, 1183 (1979).

    Google Scholar 

  48. D. F. Escande, Phys. Rep. 121, 167 (1985).

    MathSciNet  Google Scholar 

  49. L.D. Pustylnikov, Trans. Moscow Math. Soc. 2, 1 (1978).

    Google Scholar 

  50. T. Ritz, A. Hübler, preprint.

    Google Scholar 

  51. J.D. Farmer, S. Sidorowich, Phys. Rev. Lett. 59

    Google Scholar 

  52. J.P. Crutchfield, B.S. McNamara, Complex Syst. 1, 417 (1987).

    MATH  MathSciNet  Google Scholar 

  53. J. Cremers, A. Hübler, Z. Naturforsch. 42a, 797 (1987).

    MathSciNet  Google Scholar 

  54. A. Hübler, Helv. Phys. Acta 62 343 (1989); F. Ohle, F. Dinkelacker, A. Hübler, Technical Report CCSR-90–13, Center for Complex Systems Research, Univ. of Illinois at Urbana-Champaign.

    Google Scholar 

  55. H. Whitney, Ann.Math. 37, 645 (1936); 45, 220 (1936), 45 247 (1936).

    Google Scholar 

  56. D. Broomhead, R. Jones, G.P. King, J. Phys. A 20, L563 (1987).

    MATH  MathSciNet  Google Scholar 

  57. N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, Phys. Rev. Lett. 45, 712 (1980).

    Google Scholar 

  58. F. Takens, in Dynamical systems and turbulence, edited by D.A. Rand and S. Young. ( Springer, New York, 1981 ), p. 366.

    Google Scholar 

  59. T. Eisenhammer, A. Hübler, N. Packard, J.A.S. Kelso

    Google Scholar 

  60. J. Breeden, A. Hübler, Phys. Rev. A 42, 5817 (1990); J. Breeden, F. Dinkelacker, A. Hübler, Phys. Rev. A 42, 5827 (1990).

    MathSciNet  Google Scholar 

  61. M.v. Laue, Röntgenstrahleninterferenzen, Akad. Verlagsgesellschaft, Frankfurt, 3rd Ed. (1960).

    Google Scholar 

  62. P.P. Ewald, Rev. Mod. Phys. 37, 46 (1995).

    Google Scholar 

  63. C.P. Slichter, Principles of Magnetic Resonances, Harper and Row, New York (1963).

    Google Scholar 

  64. A. Corney, Atomic and Laser Spectroscopy, Clarendon Press, Oxford (1979).

    Google Scholar 

  65. T. Eisenhammer, A. Hübler, T. Geisel, E. Löscher, Phys. Rev. A 41, 3332 (1990).

    Google Scholar 

  66. B.A.Huberman and J.P.Crutchfield, Phys. Rev. Lett. 43, 1743 (1979); D.D. Humieres, M.R. Beasley, B.A. Huberman, and A. Libchaber, Phys. Rev. A 26, 2483 (1982).

    Google Scholar 

  67. D.Ruelle, Phys. Rev. Lett. 56, 405 (1986); U.Parlitz and W. Lauterborn, Phys. Lett. 107A, 351 (1986).

    Google Scholar 

  68. G. Reiser, A. Hübler, E. Löscher, Z. Naturforsch. 42a, 803 (1987).

    MathSciNet  Google Scholar 

  69. M.R. Spiegel, Schaum’s Outline of Theory and Problems of Theoretical Mechanics, McGraw-Hill, New York, 1980, p. 90.

    Google Scholar 

  70. D.N. Burghes, Modern Introduction to Classical Meachanics and Control, Halsted Press, New York, 1975.

    Google Scholar 

  71. A.D. Davies, Classical Mechanics Acad. Press, Orlando, 1986.

    Google Scholar 

  72. J.J.Brophy, Basic Electronics for Scientists McGraw-Hill, Singapore, 1984.

    Google Scholar 

  73. A. Hübler, Modeling and Control of Nonlinear Systems, Ph.D. thesis at Technische Universität München (1987).

    Google Scholar 

  74. E.A. Jackson, A. Hübler, Physica D 44, 407 (1990).

    MATH  MathSciNet  Google Scholar 

  75. R. Mettin, W. Lauterborn, A. Hübler, A. Scheeline, Phys. Rev. E 51, 4065 (1995).

    Google Scholar 

  76. J.P. Eckmann, D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).

    MATH  MathSciNet  Google Scholar 

  77. C. Reyl, L. Flepp, R. Badii, E. Brun, Phys. Rev. E 47, 267 (1993).

    Google Scholar 

  78. B. Hübinger, R. Doerner, W. Martienssen, Z. Phys. B 90, 103 (1993).

    Google Scholar 

  79. B. Hübinger, R. Doerner, W. Martienssen, M. Herdering, R. Pitka, U. Dressler, Phys. Rev. E 50, 932 (1994).

    Google Scholar 

  80. F.J. Romeiras, C. Grebogi, E. Ott, W.P. Dayawansa, Physica D 58, 165 (1992).

    MATH  MathSciNet  Google Scholar 

  81. W.L. Ditto, S.N. Rauseo, M.L. Spano, Phys. Rev. Lett. 65, 3211 (1990).

    Google Scholar 

  82. S. Hayes, C. Grebogi, E. Ott, Phys. Rev. Lett. 70, 3031 (1993).

    Google Scholar 

  83. G. Chen, X. Dong, J.Circ. Sys. Comput. 3 (1993).

    Google Scholar 

  84. J. Starett, R. Tagg, Phys. Rev. Lett. 74, 1974 (1995).

    Google Scholar 

  85. V. In, W.L. Ditto, M.L. Spano, Phys. Rev. E 51, R2689 (1995).

    Google Scholar 

  86. B. Peng, V. Petrov, K. Showalter, J. Phys. Chem. 95, 4957 (1991).

    Google Scholar 

  87. V. Petrov, B. Gaspar, J. Masere, K. Showalter, Nature 361, 240 (1993).

    Google Scholar 

  88. E.R. Hunt, Phys. Rev. Lett. 67, 1953 (1991).

    Google Scholar 

  89. R. Roy, T.W. Murphy, T.D. Maier, Z. Gills, E.R. Hunt, Phys. Rev. Lett. 68, 1259 (1992).

    Google Scholar 

  90. S. Bielawski, D. Derozier, P. Glorieux, Phys. Rev. E 49, R971 (1994).

    Google Scholar 

  91. K. Pyragas, Phys. Lett. A 170, 421 (1992); K.Pyragas, Phys. Lett. A 180, 99 (1992).

    Google Scholar 

  92. T. Shinbrot, C. Grebogi, E. Ott, J.A. Yorke, Nature 363, 411 (1993).

    Google Scholar 

  93. T. Ritz, A. Schenck zu Schweinsberg, U. Dressler, B. Hübinger, R. Doerner, W. Martienssen to be published in Chaos, Solitons 61 Fractals.

    Google Scholar 

  94. U. Dressler, G. Nitsche, Phys. Rev. Lett 68, 1 (1992);

    Google Scholar 

  95. G. Nitsche, U. Dressler, Physica D 58, 153 (1992).

    MATH  MathSciNet  Google Scholar 

  96. A. Schenck zu Schweinsberg, Die Stabilisierung chaotischer Bewegungen in rekonstruierten Zustandsräumen, Diplomarbeit at Universität Frankfurt am Main, 1995.

    Google Scholar 

  97. I.B. Schwartz, I. Triandaf, Phys. Rev. A 46, 7439 (1992);

    Google Scholar 

  98. I. Triandaf, I.B.Schwartz, Phys. Rev. E 48, 718 (1993).

    Google Scholar 

  99. T.L. Caroll, I. Triandaf, I.B. Schwartz, L. Pecora, Phys. Rev. A 46, 6189 (1992).

    Google Scholar 

  100. Z. Gills, C. Iwata, R. Roy, I.B. Schwartz, I. Triandaf, Phys. Rev. Lett. 69, 3169 (1992).

    Google Scholar 

  101. V. Petrov, M.J. Crowley, K. Showalter, Phys. Rev. Lett. 72, 2955 (1994).

    Google Scholar 

  102. R.Doerner, B. Hübinger, W. Martienssen, Int. J. Bif. Chaos

    Google Scholar 

  103. U. Dressler, T. Ritz, A. Schenck zu Schweinsberg, B. Hübinger, R. Doerner, W. Martienssen, Phys. Rev. E 51, 1845 (1995).

    Google Scholar 

  104. D. Ruelle, Physica A113, 619 (1982).

    MathSciNet  Google Scholar 

  105. P. Bak, Rep. Prog. Phys. 45, 587 (1982).

    Google Scholar 

  106. P. Reichert, R. Schilling, Phys. Rev. B30, 917 (1984).

    Google Scholar 

  107. S. Martin, W. Martienssen, Z. Phys. B68, 299 (1987).

    Google Scholar 

  108. G. Fritsch, Naturwissenschaften 75, 551 (1988);

    Google Scholar 

  109. E. Löscher, G. Fritsch, G. Jacucci (Eds.), Amorphous and Liquid Metals (NATO ASI, E118, Martinus Nijhoff, Dordrecht, Boston, Lancaster, 1987 ).

    Google Scholar 

  110. G. Dellemann, A.W. Hübler, preprint.

    Google Scholar 

  111. P. Glansdorff, I. Prigogine, Thermodynamic theory of structure, stability and fluctuations ( Wiley, New York, 1971 ).

    MATH  Google Scholar 

  112. C.G. Langton, PhD. Thesis at University of Michigan, 1988.

    Google Scholar 

  113. N.H. Packard in Dynamics Patterns in Complex Systems, edited by J.A.S. Kelso, A.J. Mandell, M.F. Schlesinger, ( World Scientific, Singapore, 1988 ), p. 293.

    Google Scholar 

  114. S.A. Kauffman, R.G. Smith, Physica D 22, 68 (1988).

    MathSciNet  Google Scholar 

  115. K.J. Kaneko, J.Suzuki at Artificial Life Conference, (Santa Fe, June 1992 ).

    Google Scholar 

  116. J.P. Crutchfield, K. Young, Phys. Rev. Lett. 63, 105 (1989).

    MathSciNet  Google Scholar 

  117. G. Kirchhoff, Collected Works, (J. Barth, Leipzig, 1882), Ch. 19, P. 232.

    Google Scholar 

  118. H. Aref, S.W. Jones, Phys. Fluids A 5, 3026 (1993).

    MATH  MathSciNet  Google Scholar 

  119. A. Galper, T. Miloh, J. Fluid Mech. 295, 91 (1995).

    MATH  MathSciNet  Google Scholar 

  120. P.D. Lax in Philadelphia SIAM Regional Conf. Ser. in April. Math.,(No. 11, 1973)

    Google Scholar 

  121. C. Wagner, H. Nosaka, K. Tsjui, A. Hiibler, preprint.

    Google Scholar 

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  1. University of Illinois at Urbana-Champaign, Urbana, IL, USA

    T. Ritz & A. W. Hübler

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  1. T. Ritz
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  1. DLR Institute for Fluid Mechanics, Germany

    G. E. A. Meier

  2. University of Karlsruhe, Germany

    G. H. Schnerr

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Ritz, T., Hübler, A.W. (1996). Optimal Control and Other Complex Systems Paradigms in the Context of Turbulent Flows. In: Meier, G.E.A., Schnerr, G.H. (eds) Control of Flow Instabilities and Unsteady Flows. International Centre for Mechanical Sciences, vol 369. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2688-2_2

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