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Spatiotemporal analysis of complex signals: Theory and applications. (English) Zbl 0943.37510

Summary: We present a space-time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos. This permits the introduction of several characteristics of the signal, three characteristic energies and entropies (one temporal, one spatial, and one global), and a characteristic dimension. Although the technique is general, we concentrate on its applications to hydrodynamic problems, specifically the transition to turbulence. We consider two cases of application: a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk. In the latter, we show a direct relation between the global entropy and the different instabilities that the flow undergoes as Reynolds number increases.

MSC:

37J60 Nonholonomic dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N99 Applications of dynamical systems
76F20 Dynamical systems approach to turbulence

Software:

Chronos
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[1] R. J. Adrian,Phys. Fluids 22:2065 (1979). · Zbl 0414.76038 · doi:10.1063/1.862515
[2] R. J. Adrian,Appl. Opt. 23:1690 (1984). · doi:10.1364/AO.23.001690
[3] N. Aubry, M. P. Chauve, and R. Guyonnet, Analysis of a rotating disk flow experiment, Preprint, B. Levich Institute, CCNY of CUNY, New York, New York (1990).
[4] N. Aubry, P. Holmes, J. L. Lumley, and E. Stone,J. Fluid Mech. 192:115 (1988). · Zbl 0643.76066 · doi:10.1017/S0022112088001818
[5] N. Aubry and S. Sanghi, inOrganized Structures and Turbulence in Fluid Mechanics, M. Lesieur, ed. (Kluwer Academic, 1989).
[6] V. I. Arnold,Bifurcations and Singularities in Mathematics and Mechanics, Theoretical and Applied Mechanics, P. Germain, M. Piau, and D. Caillerie, eds. (Elsevier, 1989).
[7] A. V. Babin and M. I. Vishic,Uspekhi Mat. Nauk 38:133 (1983) [Russ. Math. Surv. 38:151 (1983)].
[8] L. Batiston, L. Bunimovich, and R. Lima, Robustness of quasi-homogeneous configurations in coupled map lattice, Preprint, Institute for Scientific Interchange, Turin, Italy (1990).
[9] P. Bergé,Nucl. Phys. B 2:247 (1987). · doi:10.1016/0920-5632(87)90021-1
[10] P. Bergé, M. Dubois, P. Manneville, and Y. Pomeau,J. Phys. Lett. (Paris)41:L341 (1980).
[11] R. F. Blackwelder and R. E. Kaplan,J. Fluid Mech. 76:89 (1976). · doi:10.1017/S0022112076003145
[12] W. B. Brown, inBoundary Layer and Flow Control, G. V. Lachmann, ed. (Pergamon Press, 1961), p. 913.
[13] L. Bunimovich,Sou. J. Theor. Exp. Phys. 89:4 (1985).
[14] L. Bunimovich, A. Lambert, and R. Lima,J. Stat. Phys. 61 (1990).
[15] L. Bunimovich and Ya. G. Sinai,Nonlinearity 1:491-516 (1988). · Zbl 0679.58028 · doi:10.1088/0951-7715/1/4/001
[16] B. Cantwell,Annu. Rev. Fluid Mech. 13:453 (1981). · doi:10.1146/annurev.fl.13.010181.002325
[17] H. Chaté and P. Manneville,C. R. Acad. Sci. 304:609 (1987);Phys. Rev. A 38: 4351 (1988);Physica D 32:409 (1988).
[18] S. Ciliberto, F. Francini, and F. Simonelli,Opt. Commun. 54:251 (1985). · doi:10.1016/0030-4018(85)90348-7
[19] S. Ciliberto and P. Bigazzi,Phys. Rev. Rev. 60:286 (1988).
[20] S. Ciliberto and B. Nicolaenko, Estimating the number of degrees of freedom in spatially extended systems, Preprint, Instituto Nazionale di Ottica, Largo Enrico Fermi 6, 50125 Arcetri-Firenze, Italy (1990).
[21] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai,Ergodic Theory (Springer, 1980).
[22] J. Dixmier,Les Algèbres d’Opérateurs de l’Espace Hilbertien (Algèbre de von Neumann) (Gauthiers-Villars, 1968).
[23] M. J. Feigenbaum,J. Stat. Phys. 19:25 (1978). · Zbl 0509.58037 · doi:10.1007/BF01020332
[24] A. Fincham and R. Blackwelder,Bull. Am. Phys. Soc. (42nd Annu. Mtg. Div. Fluid Dynam.)1989:2266.
[25] C. Foias, G. R. Sell, and R. Témam,J. Differential Equations 73:309-353 (1988). · Zbl 0643.58004 · doi:10.1016/0022-0396(88)90110-6
[26] M. N. Glauser, S. J. Leib, and W. K. George,Turbulent Shear Flows 5 (Springer-Verlag, 1987).
[27] B. Gnedenko,The Theory of Probability (MIR, Moscow, 1976). · Zbl 0191.46702
[28] J. P. Gollub and H. L. Swinney,Phys. Rev. Lett. 35:927 (1975). · doi:10.1103/PhysRevLett.35.927
[29] P. Grassberger and I. Procaccia,Physica 9D:189 (1983).
[30] N. Gregory, J. T. Stuart, and W. S. Walker,Phil. Trans. 248:155 (1955). · Zbl 0064.43601 · doi:10.1098/rsta.1955.0013
[31] J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983). · Zbl 0515.34001
[32] J. C. R. Hunt,Trans. Can. Soc. Mech. Eng. 11:21 (1987).
[33] A. K. M. F. Hussain,J. Fluid Mech. 173:303 (1986). · doi:10.1017/S0022112086001192
[34] K. Kaneko,Physica 34D:1 (1989).
[35] J. L. Kaplan and J. A. Yorke, inFunctional Differential Equations and Approximations of Fixed Points, H. O. Peitgen and H. O. Walther, eds. (Springer, Berlin, 1979), p. 204.
[36] K. Karhunen,Ann. Acad. Sci. Fenn. Al., Math. Phys. 37:1 (1946).
[37] T. Kato,Perturbation Theory for Linear Operators (Springer-Verlag, 1966). · Zbl 0148.12601
[38] D. Keller and J. D. Farmer,Physica 23D:842 (1986).
[39] B. Khalighi,Exp. Fluids 7(2):142 (1989). · doi:10.1007/BF00207308
[40] R. Kobayashi, Y. Kohama, and Ch. Takamadate,Acta Mech. 35:71 (1980). · Zbl 0425.76029 · doi:10.1007/BF01190058
[41] A. N. Kolmogorov,Dokl. Akad. Nauk SSSR 30:301 (1941).
[42] A. Libchaber, C. Laroche, and S. Fauve,J. Phys. Lett. (Paris)43:L211 (1982).
[43] H. W. Liepmann and R. Narisimha, eds.,Turbulence Management and Relaminarisation (Springer-Verlag, 1987).
[44] M. Loève,Probability Theory (Van Nostrand, 1955).
[45] J. L. Lumley, inAtmosdpheric Turbulence and Radio Wave Propagation, A. M. Yaglom and V4. I. Tatarski, eds. (Nauka, Moscow, 1967), p. 166.
[46] J. L. Lumley,Stochastic Tools in Turbulence (Academic, Press, 1972). · Zbl 0229.76046
[47] J. L. Lumley, inTransition and Turbulence, R. E. Meyer, ed. (Academic Press, 1981), p. 215.
[48] J. L. Lumley, inWhither Turbulance?, J. L. Lumley, ed. (Springer-Verlag, 1990), p. 49.
[49] M. R. Malik, S. P. Wilkinson, and S. A. Orszag,AIAA J. 19:1131 (1981). · doi:10.2514/3.7849
[50] Mallet-Paret,J. Differential Equations 22 (1976).
[51] R. Mañe,Lecture Notes in Mathematics, Vol. 898 (Springer, 1981).
[52] J. Marsden,Butt. AMS 79:537 (1973). · Zbl 0262.76031 · doi:10.1090/S0002-9904-1973-13191-X
[53] S. E. Newhouse, D. Ruelle, and F. Takens,Commun. Math. Phys. 64:35 (1978). · Zbl 0396.58029 · doi:10.1007/BF01940759
[54] Y. Pomeau and P. Manneville,Commun. Math. Phys. 101:189 (1980). · doi:10.1007/BF01197757
[55] Y. Pomeau,Physica D 23:3 (1986). · doi:10.1016/0167-2789(86)90104-1
[56] A. I. Rakhmanov and N. K. Rakhmanova, On one dynamical system with spatial interactions, Preprint, Keldysk Institute for Applied Mathematics, Moscow (1990). · Zbl 0711.34056
[57] J. D. Rodriguez and L. Sirovich,Physica D 43:77-86 (1990). · Zbl 0723.76042 · doi:10.1016/0167-2789(90)90017-J
[58] A. Roshko,AIAA J. 14:1344 (1976).
[59] D. Ruelle,Chaotic Evolution and Strange Attractors (Cambridge University Press, 1989). · Zbl 0683.58001
[60] D. Ruelle and F. Takens,Commun. Math. Phys. 20:176 (1971). · Zbl 0223.76041 · doi:10.1007/BF01646553
[61] L. P. Silnikov,Sov. Math. Dokl. 6:163-166 (1965);Math. USSR Sbornik 6:427-438 (1968),10:91 (1970).
[62] L. Sirovich,Q. Appl. Math. 45:561-590 (1987).
[63] L. Sirovich, inProceedings 1989 Newport Conference on Turbulence (Springer-Verlag).
[64] L. Sirovich and A. E. Deane, A computational study of Rayleigh-Bénard convection. Part2: Dimension considerations, Preprint, Brown University Center for Fluid Mechanics, Providence, Rhode Island.
[65] C. R. Smith and R. D. Paxton,Exp. Fluids 1:43 (1990). · doi:10.1007/BF00282266
[66] N. H. Smith, NACA Tech. Note No. 1227 (1947).
[67] F. Takens,Lecture Notes in Mathematics (Springer-Verlag, 1981), p. 898. · Zbl 0513.58032
[68] R. Témam,Infinite Dimensional Dynamical Systems in Mechanics and Physics (Springer-Verlag, New York, 1988).
[69] H. Tennekes and J. L. Lumley,A First Course in Turbulence (MIT Press). · Zbl 0285.76018
[70] I. Waller and R. Kapral,Phys. Rev. 30A:2047 (1984).
[71] J. M. Wallace and F. Hussain,Appl. Mech. Rev. 43:S203 (1990). · doi:10.1115/1.3120807
[72] W. W. Willmarth, inAdvances in Applied Mechanics 15 (Academic Press, 1975), p. 159.
[73] I. Yamashita and M. Takematsu,Rep. Inst. Appl. Mech. Hyushu Univ. (Japan)22(69) (1974).
[74] S. Ciliberto and M. Caponeri,Phys. Rev. Lett. 1990:2775-2778.
[75] S. Ciliberto, inProceedings Les Houches, Complexity and Dynamics (1990), to appear.
[76] A. E. Deane, I. G. Kevrekidis, G. E. Karniadakis, and S. A. Orszag, Low dimensional models for complex flows geometry flows: Application to grooved channels and circular cylinders, Preprint (1990). · Zbl 0746.76021
[77] M. Kirby, D. Armbruster, and W. Güttinger, An approach for the analysis of spatially localized oscillations, inConference Proceedings: Bifurcations and Chaos, Würzburg (to appear).
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