[1] R. J., Adrian. Conditional eddies in isotropic turbulence. Physics of Fluids, 22(11):2065–70, 1979.

[2] R. J., Adrian and P., Moin. Stochastic estimation of organized turbulent structure: Homogeneous shear flow. J. Fluid Mech., 190:531–59, 1988.

[3] R. J., Adrian, P., Moin, and R. D., Moser. Stochastic estimation of conditional eddies in turbulent channel flow. In CTR, Proceedings of the 1987 Summer Program, Stanford University, CA, 1987. Center for Turbulence Research.Google Scholar

[4] S., AhujaandC. W., Rowley. Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. J. FluidMech., 645:447–478, Feb. 2010.

[5] V. R., Algazi and D. J., Sakrison. On the optimality of the Karhunen-Loève expansion. IEEE Trans. Inform. Theory, 15:319–21, 1969.

[6] C. D., Andereck, S. S., Liu, and H., Swinney. Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech., 164:155–83, 1986.

[7] C. A., Andrews, J. M., Davies, and G. R., Schwartz. Adaptive data compression. Proc. IEEE, 55:267–77, 1967.

[8] A. A., Andronov, E. A., Vitt, and S. E., Khaikin. Theory of Oscillators. Pergamon Press, Oxford, UK, 1966. Reprinted 1987 by Dover Publications, New York.Google Scholar

[9] D., Armbruster, J., Guckenheimer, and P., Holmes. Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry. Physica D, 29:257–82, 1988.

[10] D., Armbruster, J., Guckenheimer, and P., Holmes. Kuramoto-Sivashinsky dynamics on the center unstable manifold. SIAM J. on Appl. Math., 49:676–91, 1989.

[11] D., Armbruster, R., Heiland, and E. J., Kostelich. KLTOOL: a tool to analyze spatio-temporal complexity. Chaos, 4(2):421, 1994.

[12] D., Armbruster, R., Heiland, E. J., Kostelich, and B., Nicolaenko. Phase-space analysis of bursting behavior in Kolmogorov flow. Physica D, 58:392–401, 1992.

[13] D., Armbruster, B., Nicolaenko, N., Smaoui, and P., Chossat. Analysing bifurcations in the Kolmogorov flow equations. In P., Chossat, editor, Dynamics, Bifurcations and Symmetries, pages 11–33, Dordrecht, Kluwer, 1994.Google Scholar

[14] L., Arnold. Stochastic Differential Equations. John Wiley, New York, 1974.Google Scholar

[15] V. I., Arnold. Ordinary Differential Equations. MIT Press, Cambridge, MA, 1973.Google Scholar

[16] V. I., Arnold. Mathematical Methods of Classical Mechanics.Springer-Verlag, New York, 1978.Google Scholar

[17] V. I., Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, New York, 1983.Google Scholar

[18] D. K., Arrowsmith and C. M., Place. An Introduction to Dynamical Systems. Cambridge University Press, Cambridge, UK, 1990.Google Scholar

[19] N., Aubry. A Dynamical System/Coherent Structure Approach to the Fully Developed Turbulent Wall Layer. PhD thesis, Cornell University, 1987.Google Scholar

[20] N., Aubry, R., Guyonnet, and R., Lima. Spatio-temporal analysis of complex signals: theory and applications. J. Stat. Phys., 64(3/4):683-739, 1991.Google Scholar

[21] N., Aubry, R., Guyonnet, and R., Lima. Spatio-temporal symmetries and bifurcations via bi-orthogonal decompositions. J. Nonlinear Sci., 2:183–215, 1992.

[22] N., Aubry, P., Holmes, J. L., Lumley, and E., Stone. The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech., 192:115–73, 1988.

[23] N., Aubry, W.-L., Lian, and E. S., Titi. Preserving symmetries in the proper orthogonal decomposition. SIAM J. onSci. Comput., 14:483–505, 1993.

[24] N., Aubry, J. L., Lumley, and P., Holmes. The effect of modeled drag reduction in the wall region. Theoret. Comput. Fluid Dynamics, 1:229∧-8, 1990.Google Scholar

[25] N., Aubry and S., Sanghi. Bifurcations and bursting of streaks in the turbulent wall layer. In M., Lesieur and O., Métais, editors, Turbulence and Coherent Structures, pages 227–51, Dordrecht, Kluwer, 1991.Google Scholar

[26] A., Back, J., Guckenheimer, M., Myers, F. J., Wicklin, and P., Worfolk. DsTool: Computer assisted exploration of dynamical systems. AMS Notices, April 1992, 39(4):303–9, 1992.

[27] S., Bagheri, J., Hœpffner, P. J., Schmid, and D. S., Henningson. Input-output analysis and control design applied to a linear model of spatially developing flows. Applied Mechanics Reviews, 62(2):020803, Mar. 2009.

[28] P., Bakewell and J. L., Lumley. Viscous sublayer and adjacent wall region in turbulent pipe flows. Physics of Fluids, 10:1880–9, 1967.

[29] K. S., Ball, L., Sirovich, and L. R., Keefe. Dynamical eigenfunction decomposition of turbulent channel flow. In T. J. for Num. Meth. in Fluids, 12:585–604, 1991.

[30] P. R., Bandyopadhyay, editor. Application of Microfabrication to Fluid Mechanics 1994, volume FED-Vol.197. Amer. Soc. Mech. Eng., New York, 1994.

[31] H., Bartlet, A. W., Lohmann, and B., Wirnitzer. Phase and amplitude recovery from bispectra. Appl. Opt., 23:3121–9, 1984.

[32] G. K., Batchelor. The Theory of Homogeneous Turbulence.Cambridge University Press, Cambridge, UK, 1956.Google Scholar

[33] G. K., Batchelor and I., Proudman. The large scale structure of homogeneous turbulence. Phil. Trans. Roy. Soc., A248(949):369–405, 1956.

[34] M., Bergmann and L., Cordier. Optimal control of the cylinder wake in the laminar regime by Trust-Region methods and POD reduced order models. J. Comp. Phys., 227:7813–40, 2008.

[35] M., Bergmann, L., Cordier, and J.-P., Brancher. Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced order model. Physics of Fluids, 17:097101, 2005.Google Scholar

[36] G., Berkooz. A numerical evaluation of the dynamical systems approach to wall layer turbulence. In CTR, Proceedings of the 1990 Summer Program, Stanford University, CA, 1990. Center for Turbulence Research.Google Scholar

[37] G., Berkooz. Turbulence, Coherent Structures, and Low Dimensional Models. PhD thesis, Cornell University, 1991.Google Scholar

[38] G., Berkooz. Controlling models of the turbulent wall layer by boundary deformation. Technical Report FDA-92-16, Cornell University, 1993.

[39] G., Berkooz. An observation on probability density equations, or, When do simulations reproduce statistics? Nonlinearity, 7:313–28, 1994.

[40] G., Berkooz, L. P., Chew, R., Palmer, and R., Zippel. Generating spectral methods solvers for partial differential equations. In IMACS International Conference on Computational Methods for Partial Differential Equations, pages 471–6, New Brunswick, NJ, June 1992.Google Scholar

[41] G., Berkooz, T., Corke, M. N., Glauser, M., Psiaki, and M., Fisher. Design for control of flow instabilities: first principles and an application. In J., Paduano, editor, Sensing, Actuation, and Control in Aeropropulsion. SPIE, 1995.Google Scholar

[42] G., Berkooz, J., Elezgaray, and P., Holmes. Coherent structures in random media and wavelets. Physica D, 61:47–58, 1992.

[43] G., Berkooz, P., Holmes, and J. L., Lumley. Intermittent dynamics in simple models of the wall layer. J. Fluid Mech., 230:75–95, 1991.

[44] G., Berkooz, P., Holmes, and J. L., Lumley. On the relation between low dimensional models and the dynamics of coherent structures in the turbulent wall layer. Theoret. Comput. Fluid Dynamics, 4:255–69, 1993.

[45] G., Berkooz, P., Holmes, and J. L., Lumley. The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech., 25:539–75, 1993.

[46] G., Berkooz, P., Holmes, J. L., Lumley, N., Aubry, and E., Stone. Observations regarding “Coherence and chaos in a model of turbulent boundary layer” by X., Zhou and L., Sirovich. Physics of Fluids A, 6:1574–8, 1994.

[47] G., Berkooz, M., Psiaki, and M., Fisher. Estimation and control of models of the turbulent wall layer. In Active Control of Noise and Vibration, New York, 1995. ASME.Google Scholar

[48] G., Berkooz and E. S., Titi. Galerkin projections and the proper orthogonal decomposition for equivariant equations. Phys. Lett. A, 174:94–102, 1993.

[49] L. P., Bernal, R. E., Breidenthal, G. L., Brown, J. H., Konrad, and A., Roshko. On the development of three dimensional small scales in turbulent mixing layers. In Second Symposium on Turbulent Shear Flows, pages 8.1–8.6, 1979.Google Scholar

[50] W.-J., Beyn and V., Thümmler. Freezing solutions of equivariant evolution equations. SIAM Journal on Applied Dynamical Systems, 3(2): 85-116, 2004.Google Scholar

[51] G. D., Birkhoff. Dynamical Systems. American Mathematical Society, Providence, RI, 1927.Google Scholar

[52] R. F., Blackwelder and R. E., Kaplan. On the wall structure of the turbulent boundary layer. J. Fluid Mech., 76:89–112, 1976.

[53] D. G., Bogard and W. G., Tiederman. Burst detection with single point velocity measurements. J. FluidMech., 162:38913, 1986.Google Scholar

[54] J. P., Bonnet, D. R., Cole, J., Delville, M. N., Glauser, and L., Ukeiley. Stochastic estimation and proper orthogonal decomposition: complementary techniques for identifying structure. Experiments in Fluids, 17:307–14, 1994.

[55] R., Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, volume 470 of Springer Lecture Notes in Mathematics. Springer-Verlag, New York, 1975.Google Scholar

[56] W. E., Boyce and R. C., DiPrima. Elementary Differential Equations and Boundary Value Problems. Wiley, New York, fifth edition, 1992.Google Scholar

[57] L., Breiman. Probability. Addison Wesley, Reading, MA, 1968.Google Scholar

[58] J., Bridges, H. S., Husain, and F., Hussain. Whither coherent structures? Comment 1. In J. L., Lumley, editor, Whither Turbulence? Turbulence at the Crossroads, pages 132–51. Springer-Verlag, New York, 1990.Google Scholar

[59] D. R., Brillinger and M., Rosenblatt. Asymptotic theory of estimates of kth order spectra. In B., Harris, editor, Spectral Analysis of Time Series. Wiley, New York, 1967.Google Scholar

[60] G. L., Brown and A., Roshko. The effect of density difference on the turbulent mixing layer. In A.G.A.R.D. Conference on Turbulent Shear Flows, Conf. Proceedings No. 93, pages 23.1–23.12. NATO Advisory Group for Aerospace Research and Development, 1971.Google Scholar

[61] G. L., Brown and A., Roshko. On density effects and large structure in turbulent mixing layers. J. Fluid Mech., 64:775–816, 1974.

[62] A. D., Bruno. Local Methods in Nonlinear Differential Equations. Springer-Verlag, New York, 1989.Google Scholar

[63] V., Brunsden, J., Cortell, and P., Holmes. Power spectra of chaotic oscillations of a buckled beam. J. Sound and Vibration, 130:1–25, 1989.

[64] V., Brunsden and P., Holmes. Power spectra of strange attractors near homoclinic orbits. Phys. Rev. Lett., 58:1699–702, 1987.

[65] J. M., Burgers. A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech, 1:171–99, 1948.

[66] R M., Busse and R. M., Clever. Nonstationary convection in a rotating system. In U., Muller, K. G., Roessner, and B., Schmidt, editors, Recent Developments in Theoretical and Experimental Fluid Mechanics, pages 376–85. Springer-Verlag, New York, 1979.Google Scholar

[67] R M., Busse and K. E., Heikes. Convection in a rotating layer: a simple case of turbulence. Science, 208:173–5, 1980.

[68] S. A., Campbell and P., Holmes. Bifurcation from O(2) symmetric heteroclinic cycles with three interacting modes. Nonlinearity, 4:697–726, 1991.

[69] S. A., Campbell and P., Holmes. Heteroclinic cycles and modulated travelling waves in a system with D(4) symmetry. Physica D, 59:52–78, 1992.

[70] C., Canuto, M. Y., Hussaini, A., Quarteroni, and T. A., Zang. Spectral Methods in Fluid Dynamics. Springer-Verlag, New York, 1988.Google Scholar

[71] J., Carr. Applications of Center Manifold Theory. Springer-Verlag, New York, 1981.Google Scholar

[72] D. H., Chambers, R. J., Adrian, P., Moin, D. S., Stewart, and H. J., Sung. Karhunen-Loève expansion of Burgers model of turbulence. Phys. Fluids, 31:2573–82, 1988.

[73] J., Chomaz. Global instabilities in spatially developing flows: Non-normality and nonlinearity. Annual Review of Fluid Mechanics, 37:357–92, 2005.

[74] J. M., Chomaz, P., Huerre, and L. G., Redekopp. Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett., 60(1):25–8, 1988.

[75] P., Chossat and G., Iooss. Primary and secondary bifurcations in the Couette-Taylor problem. Japan J. Appl. Math., 2:37–68, 1985.

[76] P., Chossat and G., Iooss. The Couette-Taylor Problem. Springer-Verlag, New York, 1994.Google Scholar

[77] E. A., Coddington and N., Levinson. Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.Google Scholar

[78] B. D., Coller and P., Holmes. Suppression of bursting. Automatica, 33(1):1–11, 1997.

[79] B. D., Coller, P., Holmes, and J. L., Lumley. Control of bursting in boundary layer models. Appl. Mech. Rev., 47(6):part 2, S139-43, 1994.Google Scholar

[80] B. D., Coller, P., Holmes, and J. L., Lumley. Controlling noisy heteroclinic cycles. Physica D, 72:135–60, 1994.

[81] B. D., Coller, P., Holmes, and J. L., Lumley. Interaction of adjacent bursts in the wall region. Phys. Fluids, 6(2):954–61, 1994.

[82] P., Collet and J.-P. Eckmann. Iterated Maps on the Interval as Dynamical Systems. Birkhauser, Basel, 1980.Google Scholar

[83] P., Constantin and C., Foias. Navier-Stokes equations. Chicago University Press, Chicago, IL, 1988.Google Scholar

[84] P., Constantin, C., Foias, and R., Temam. Attractors Representing Turbulent Flows. Memoirs of the AMS 53, 314. American Mathematical Society, Providence, RI, 1985.Google Scholar

[85] P., Constantin, C., Foias, R., Temam, and B., Nicolaenko. Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Springer-Verlag, New York, 1989.Google Scholar

[86] I. P., Cornfeld, S. V, Fomin, and Y G., Sinai. Ergodic Theory.Springer-Verlag, New York, 1982.Google Scholar

[87] S., Corrsin and A. L., Kistler. The Free-stream Boundaries of Turbulent Flows. NACA TN 3133. National Advisory Committee for Aeronautics, 1954.Google Scholar

[88] C., Cossu and J. M., Chomaz. Global measures of local convective instabilities. Phys. Rev. Lett., 78(23):4387–1390, June 1997.

[89] H., Dankowicz, P., Holmes, G., Berkooz, and J., Elezgaray. Local models of spatio-temporally complex fields. Physica D, 90:387–407, 1996.

[90] I., Daubechies. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, 61. SIAM Publications, Philadelphia, PA, 1992.Google Scholar

[91] A. E., Deane, I. G., Kevrekidis, G. E., Karniadakis, and S. A., Orszag. Low-dimensional models for complex flows: Application to grooved channels and circular cylinders. Physics of Fluids A, 3(10):2337–54, 1991.

[92] A. E., Deane and L., Sirovich. A computational study of Rayleigh-Bénard convection. Part I. Rayleigh number scaling. J. Fluid Mech., 222:231–50, 1991.

[93] J. W., Deardorff. A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech., 41:453–80, 1970.

[94] R. G., Deissler. Is Navier-Stokes turbulence chaotic? Physics of Fluids, 29(5):1453–7, 1986.

[95] Y., Demay, G., Iooss, and P., Laure. Wave patterns in small gap Couette-Taylor problem. European J. Mech. B.|Fluids, 11(5):621–34, 1992.

[96] C. R., Doering and J. D., Gibbon. Applied Analysis of the Navier-Stokes Equations. Cambridge University Press, Cambridge, UK, 1994.Google Scholar

[97] P. G., Drazin and W. H., Reid. Hydrodynamic Stability. Cambridge University Press, Cambridge, UK, 1981.Google Scholar

[98] G. E., Dullerud and F., Paganini. A Course in Robust Control Theory: A Convex Approach, volume 36 of Texts in Applied Mathematics. Springer-Verlag, 1999.Google Scholar

[99] F., Dyson. The scientist as rebel. New York Review of Books, XLII(9):31-3, May 25, 1995.Google Scholar

[100] B., Eckhardt and A., Mersmann. Transition to turbulence in a shear flow. Phys. Rev. E, 60:509–17, 1999.

[101] J., Elezgaray, G., Berkooz, and P., Holmes. Wavelet analysis of the motion of coherent structures. In Y., Meyer and S., Roques, editors, Progress in Wavelet Analysis and Applications, pages 471–6, Gif-sur-Yvette, France, 1993. Editions Frontières.Google Scholar

[102] H., Faisst. The transition from the Taylor-Couette system to the plane Couette system. Diplomarbeit, Universität Marburg, 1999.

[103] H., Faisst and B., Eckhardt. Transition from the Couette-Taylor system to the plane Couette system. Phys. Rev. E, 61:7227–30, 2000.

[104] M., Farge. Wavelet transforms and their applications to turbulence. Ann. Rev. Fluid Mech. 24:395–457, 1992.

[105] B. F., Farrell and P. J., Ioannou. Stochastic forcing of the linearized Navier-Stokes equations. Physics of Fluids A, 5(11):2600–9, 1993.

[106] B. F., Farrell and P. J., Ioannou. A theory for the statistical equilibrium energy spectrum and heat flux produced by transient baroclinic waves. J. Atmos. Sci., 51(19):2685–98, 1994.

[107] B. F., Farrell and P. J., Ioannou. Variance maintained by stochastic forcing of non-normal dynamical systems associated with linearly stable shear flows. Phys. Rev. Lett., 72(8): 1188-91, 1994.Google Scholar

[108] W., Feller. An Introduction to Probability Theory and its Applications. Wiley, New York, 1957.Google Scholar

[109] J. A., Ferre and F., Girlat. Pattern-recognition analysis of the velocity field in plane turbulent wakes. J. FluidMech., 198:27–64, 1989.

[110] M., Fogleman. Low-Dimensional Models of Internal Combustion Engine Flows using the Proper Orthogonal Decomposition. PhD thesis, Cornell University, 2005.Google Scholar

[111] M., Fogleman, J., Lumley, D., Rempfer, and D., Hawarth. Application of the proper orthogonal decomposition to datasets of internal combustion engine flows. J. Turbulence, 5:023, 2005.Google Scholar

[112] C., Foias, O., Manley, and L., Sirovich. Empirical and Stokes eigenfunctions and the far dissipative turbulent spectrum. Physics of Fluids A, 2:464–7, 1990.

[113] C., Foias, O., Manley, and R., Temam. Approximate inertial manifolds and effective viscosity in turbulent flows. Technical Report 9011, Dept. of Math., Indiana University, Bloomington, IN, 1990.

[114] C., Foias, G., Sell, and R., Temam. Inertial manifolds for nonlinear evolutionary equations. J. Diff. Eqns, 73:199–224, 1989.

[115] C., Foias, G. R., Sell, and E. S., Titi. Exponential tracking and approximation of inertial manifolds for dissipative equations. J. of Dynamics andDiff. Eqns, 1:199–224, 1989.

[116] C., Foias and R., Temam. Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Functional Anal., 87:359–69, 1989.

[117] S. K., Friedlander and L., Topper, editors. Turbulence: Classical Papers on Statistical Theory. Interscience, New York, 1962.Google Scholar

[118] D. H., Gay and W. H., Ray. Identification and control of linear distributed parameter systems through the use of experimentally determined singular functions. In Proc. IFAC Symp. Control of Distributed Parameter Systems, Los Angeles, CA, 30 June-2 July 1986, pages 173–9, 1986.Google Scholar

[119] D. H., Gay and W. H., Ray. Application of singular value methods for identification and model based control of distributed parameter systems. In Proc. IFAC Workshop on Model Based Process Control, Atlanta, GA, June 1988, pages 95–102, 1988.Google Scholar

[120] J. F., Gibson, J., Halcrow, and P., Cvitanovic. Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech., 638:243–66, 2009.

[121] M. N., Glauser and W. K., George. Orthogonal decomposition of the axisymmetric jet mixing layer including azimuthal dependence. In G. Comte-Bellot and J., Mathieu, editors, Advances in Turbulence, pages 357–66. Springer-Verlag, New York, 1987.Google Scholar

[122] M. N., Glauser and W. K., George. An orthogonal decomposition of the axisymmetric jet mixing layer utilizing cross-wire velocity measurements. In Sixth Symposium on Turbulent Shear Flows, Toulouse, France, pages 10.1.1–10.1.6, Toulouse, France, 1987. Ecole Nationale Superieure de l'Aeronautique et de l'Espace and ONERA Centre d'Etudes et de Recherches de Toulouse.Google Scholar

[123] M. N., Glauser, S. J., Leib, and W. K., George. Coherent structures in the axisymmetric turbulent jet mixing layer. In F., Durst, B. E., Launder, J. L., Lumley, F. W., Schmidt, and J., Whitelaw, editors, Turbulent Shear Flows 5, pages 134—45. Springer-Verlag, New York, 1987.Google Scholar

[124] M. N., Glauser and X., Zheng. A low-dimensional dynamical systems description of the axisymmetric jet mixing layer. Technical Report MAE-247, Clarkson University, 1991.

[125] M. N., Glauser, X., Zheng, and C. R., Doering. The dynamics of organized structures in the axisymmetric jet mixing layer. In M., Lesieur and O., Métais, editors, Turbulence and Coherent Structures, pages 253–65. Kluwer, Dordrecht, 1991.Google Scholar

[126] M. N., Glauser, X., Zheng, and W. K., George. The streamwise evolution of coherent structures in the axisymmetric jet mixing layer. In T. B., Gatski, S., Sarkar, and C. G., Speziale, editors, Studies in Turbulence, pages 207–22. Springer-Verlag, New York, 1992.Google Scholar

[127] S., Glavaški, J. E., Marsden, and R. M., Murray. Model reduction, centering, and the Karhunen- Loève expansion. In Proc. IEEE Conf. Decision and Control, volume 37, pages 2071–2076, 1998.Google Scholar

[128] J., Gleick. Chaos: Making a New Science. Viking Penguin, Inc., New York, 1987.Google Scholar

[129] P., Glendinning. Stability, Instability and Chaos.Cambridge University Press, Cambridge, UK, 1995.Google Scholar

[130] A., Glezer, A. J., Kadioglu, and A. J., Pearlstein. Development of an extended proper orthogonal decomposition and its application to a time periodically forced plane mixing layer. Physics of Fluids A, 1:1363–73, 1989.

[131] M. A., Gol'dshtik. Structural turbulence. Institute of Thermophysics, Novosibirsk, USSR, 1982.Google Scholar

[132] S., Goldstein. Modern Developments in Fluid Dynamics. Oxford University Press, Oxford, UK, 1952.Google Scholar

[133] M., Golubitsky and W. F., Langford. Pattern formation and bistability in flow between counterrotating cylinders. Physica D, 32:362–92, 1988.

[134] M., Golubitsky and D. G., Schaeffer. Singularities and Groups in Bifurcation Theory, Volume I. Springer-Verlag, New York, 1985.Google Scholar

[135] M., Golubitsky and I., Stewart. Symmetry and stability in Taylor-Couette flow. SIAM J. on Math. Anal., 17(2):249–88, 1986.

[136] M., Golubitsky, I., Stewart, and D. G., Schaeffer. Singularities and Groups in Bifurcation Theory, Volume II. Springer-Verlag, New York, 1988.Google Scholar

[137] M., Gorman and P. J., Widmann. Nonlinear dynamics of a convection loop: a quantitative comparison of experiments with theory. Physica D, 19:255–67, 1986.

[138] D., Gottlieb and S. A., Orszag. Numerical Analysis of Spectral Methods: Theory and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, 26. SIAM Publications, Philadelphia, PA, 1977.Google Scholar

[139] W. R., Graham, J., Peraire, and K. Y., Tang. Optimal control of vortex shedding using low-order models. Part I: Open-loop model development. In T. J. Numer. Meth. Engrng, 44:945–72, 1999.

[140] C., Grebogi, S. M., Hammel, J. A., Yorke, and T., Sauer. Shadowing of physical trajectories in chaotic dynamics: containment and refinement. Phys. Rev. Lett., 65(13):1527–30, 1990.

[141] R R, Grinstein, M. N., Glauser, and W. K., George. A low-dimensional dynamical systems description of coherent structures in the axisymmetric jet mixing layer. In S. I., Green, editor, Fluid Vortices, pages 65–94. Kluwer, Dordrecht, 1995.Google Scholar

[142] J., Guckenheimer. A strange, strange attractor. In J. E., Marsden and M., McCracken, editors, The Hopf Bifurcation and its Applications, pages 368–81. Springer-Verlag, New York, 1976.Google Scholar

[143] J., Guckenheimer. The role of geometry in computational dynamics. In P., Chossat, editor, Dynamics, Bifurcations and Symmetries, pages 155–66. Kluwer, Dordrecht, 1994.Google Scholar

[144] J., Guckenheimer and P., Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983.Google Scholar

[145] J., Guckenheimer and P., Holmes. Structurally stable heteroclinic cycles. Math. Proc. Cambridge Phil. Soc., 103:189–92, 1988.

[146] J., Guckenheimer and A., Mahalov. Resonant triad interactions in symmetric systems. Physica D, 54:267–310, 1992.

[147] J., Guckenheimer and R. R Williams. Structural stability of Lorenz attractors. Publ. Math. IHES, 50:59–72, 1979.

[148] J., Halcrow, J. R Gibson, and P., Cvitanovic. Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech., 611:107–30, 2008.

[149] J., Halcrow, J. R Gibson, P., Cvitanovic, and D., Visawanth. Heteroclinic connections in plane Couette flow. J. Fluid Mech., 621:365–76, 2009.

[150] J., Hale. Ordinary Differential Equations. Wiley, New York, 1969.Google Scholar

[151] J. M., Hamilton, J., Kim, and R Waleffe. Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech., 287:317–8, 1995.Google Scholar

[152] D. C., Hawarth. Large-eddy simulation of in-cylinder flows. Oil and Gas Science and Technology, 54:175–85, 1999.

[153] D. C., Hawarth and K., Jensen. Large-eddy simulation on unstructured deforming meshes: Towards reciprocating IC engines. Computers & Fluids, 29:023, 2000.Google Scholar

[154] D., Henry. Geometric Theory of Semilinear Parabolic Equations, volume 840 of Springer Lecture Notes in Mathematics. Springer-Verlag, New York, 1980.Google Scholar

[155] S., Herzog. The Large Scale Structure in the Near Wall Region of a Turbulent Pipe Flow. PhD thesis, Cornell University, 1986.Google Scholar

[156] M. W., Hirsch, S., Smale, and R. L., Devaney. Differential Equations, Dynamical Systems and an Introduction to Chaos. Academic Press/Elsevier, San Diego, CA, 2004.Google Scholar

[157] M., Högberg, T. R., Bewley, and D. S., Henningson. Linear feedback control and estimation of transition in plane channel flow. J. Fluid Mech., 481:149–75, 2003.

[158] P., Holmes. Can dynamical systems approach turbulence? In J. L., Lumley, editor, Whither Turbulence? Turbulence at the Crossroads, pages 195–249. Springer-Verlag, New York, 1990.Google Scholar

[159] P., Holmes. On Moffatt's paradox or, can empirical projections approach turbulence? In J. L., Lumley, editor, Whither Turbulence? Turbulence at the Crossroads, pages 306–9. Springer- Verlag, New York, 1990.Google Scholar

[160] P., Holmes. Symmetries, heteroclinic cycles and intermittency in fluid flow. In G. R., Sell, C., Foias, and R., Temam, editors, Turbulence in Fluid Flows: a Dynamical Systems Approach, pages 49–58. Springer-Verlag, New York, 1993.Google Scholar

[161] P., Holmes, G., Berkooz, and J. L., Lumley. Turbulence, dynamical systems and the unreasonable effectiveness of empirical eigenfunctions. In Proceedings of the International Congress of Mathematicians, Kyoto, 1990, pages 1607–17. Springer-Verlag, Tokyo, 1991.Google Scholar

[162] P., Holmes and E., Stone. Heteroclinic cycles, exponential tails and intermittency in turbulence production. In T. B., Gatski, S., Sarkar, and C. G., Speziale, editors, Studies in Turbulence, pages 179–89. Springer-Verlag, New York, 1992.Google Scholar

[163] E., Hopf. A mathematical example displaying the features of turbulence. Comm. Pure Appl. Math., 1:303–22, 1948.

[164] E., Hopf. On the application of functional calculus to the statistical theory of turbulence. In Proc. Symp. Appl. Math., Providence, RI, 1957, American Math. Soc.Google Scholar

[165] S., Hoyas and J., Jiménez. Scaling of the velocity fluctuations in turbulent channels up to Reτ = 2003. Physics of Fluids, 18: 011702, 2006.Google Scholar

[166] P., Huerre and P. A., Monkewitz. Local and global instabilities in spatially developing flows. Annual Review of Fluid Mechanics, 22:473–537, 1990.

[167] M., Hultmark, M., Leftwich, and A. J., Smits. Flowfield measurements in the wake of a robotic lamprey. Experiments in Fluids, 43:683–90, 2007.

[168] A. K., M. F., Hussain and W. C., Reynolds. The mechanisms of an organized wave in turbulent shear flow. J. Fluid Mech., 41:241–58, 1970.

[169] J. M., Hyman, B., Nicolaenko, and S., Zaleski. Order and complexity in the Kuramoto- Sivashinsky model of weakly turbulent interfaces. Physica D, 23:265–92, 1986.

[170] M., Ilak. Model Reduction and Feedback Control of Transitional Channel Flow. PhD thesis, Princeton University, 2009.Google Scholar

[171] M., Ilak, S., Bagheri, L., Brandt, C. W., Rowley, and D. S., Henningson. Model reduction of the nonlinear complex Ginzburg-Landau equation. SIAM Journal on Applied Dynamical Systems, 9(4): 1284-1302, 2010.Google Scholar

[172] M., Ilak and C. W., Rowley. Modeling of transitional channel flow using balanced proper orthogonal decomposition. Physics of Fluids, 20:034103, 2008.Google Scholar

[173] G., Iooss. Secondary bifurcations of Taylor vortices into wavy inflow or outflow boundaries. J. FluidMech., 173:273–88, 1986.

[174] G., Iooss and A., Mielke. Time-periodic Ginzburg-Landau equations for one dimensional patterns with large wave length. Z Angew. Math. Phys., 43:125–38, 1992.

[175] J., Jiménez and P., Moin. The minimal flow unit in near-wall turbulence. J. Fluid Mech., 225:213–40, 1991.

[176] C., Jones and M. K., Proctor. Strong spatial resonance and travelling waves in Bénard convection. Phys. Lett. A, 121:224–7, 1987.

[177] B. H., Jørgensen, J. N., Sørensen, and M. Brøns. Low-dimensional modeling of a driven cavity flow with two free parameters. Theoret. Comput. Fluid Dynamics, 16:299–317, 2003.

[178] J.-N., Juang and R. S., Pappa. An eigensystem realization algorithm for modal parameter identification and model reduction. Journal of Guidance, Control, and Dynamics, 8(5):620–7, 1985.

[179] V, Juttijudata, J. L., Lumley, and D., Rempfer. Proper orthogonal decomposition in Squire's coordinate system for dynamical models of channel turbulence. J. FluidMech., 534:195–225, 2005.

[180] Y. S., Kachanov, V. V., Kozlov, V. Y., Levchenko, and M. P., Ramazov. On the nature of K-breakdown of a laminar boundary layer. In V., Kozlov, editor, Laminar-Turbulent Transition, pages 61–73. Springer-Verlag, New York, 1985.Google Scholar

[181] B. A., Kader. Change in the thickness of an incompressible turbulent boundary layer in the presence of a longitudinal pressure gradient. Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 2:150–6, 1979. Translated in Fluid Dynamics, 14(2):283–9, 1979.

[182] B. A., Kader. Hydrodynamic structure of accelerated turbulent boundary layers. Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 3:29–37, 1983. Translated in Fluid Dynamics, 18(3):360–7, 1983.

[183] K., Karhunen. Zur Spektraltheorie stochastischer Prozesse. Ann. Acad. Sci. Fennicae, Ser. A1, 34, 1946.Google Scholar

[184] C., Kasnakoglu, A., Serrani, and M. O., Efe. Control input separation by actuation mode expansion for flow control problems. In T. J. Control, 81(9):1475–92, 2008.

[185] G., Kawahara and S., Kida. Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. FluidMech., 449:291–300, 2001.

[186] L., Keefe, P., Moin, and J., Kim. The dimension of an attractor in turbulent Poiseuille flow. Bull. Am. Phys. Soc., 32:2026, 1987.Google Scholar

[187] A., Kelley. The stable, center stable, center, center unstable and unstable manifolds. J. Diff. Eqns, 3:546–70, 1967.

[188] I. G., Kevrekidis, B., Nicolaenko, and C., Scovel. Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation. SIAM J. onAppl. Math., 50:760–90, 1990.

[189] H. T., Kim, S. J., Kline, and W. C., Reynolds. The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech., 50:133–60, 1971.

[190] J., Kim, P., Moin, and R. J., Moser. Turbulence statistics in a fully developed channel flow at low Reynolds number. J. Fluid Mech., 177:133–66, 1987.

[191] M., Kirby and D., Armbruster. Reconstructing phase space from PDE simulations. Z Angew. Math. Phys., 43:999–1022, 1992.

[192] M., Kirby, J., Boris, and L., Sirovich. An eigenfunction analysis of axisymmetric jet flow. J. of Computational Physics, 90(1):98–122, 1990.

[193] M., Kirby, J., Boris, and L., Sirovich. A proper orthogonal decomposition of a simulated supersonic shear layer. International J. for Numerical Methods in Fluids, 10:411–28, 1990.

[194] S. J., Kline. Observed structure features in turbulent and transitional boundary layers. In G., Sovran, editor, Fluid Mechanics of Internal Flow, pages 27–68, Amsterdam, 1967. Elsevier.Google Scholar

[195] S. J., Kline, W. C., Reynolds, E A., Schraub, and P. W., Runstadler. The structure of turbulent boundary layers. J. Fluid Mech., 30:741–73, 1967.

[196] J. J., Kobine and T., Mullin. Low dimensional bifurcation phenomena in Taylor-Couette flow with discrete azimuthal symmetry. J. FluidMech., 275:379–405, 1994.

[197] D. D., Kosambi. Statistics in function space. J. Indian Math. Soc., 7:76–88, 1943.

[198] M., Krupa. Robust heteroclinic cycles. Forschungsbericht 2, Inst. für Angewandte und Numerische Mathematik, TU Wein, Austria, 1994.

[199] M., Krupa and I., Melbourne. Asymptotic stability of heteroclinic cycles in systems with symmetry. Ergodic Theory and Dynamical Systems, 15:121—47, 1995.Google Scholar

[200] Y, Kuramoto. Diffusion-induced chaos in reaction systems. Suppl. Prog. Theor. Phys., 64:346- 67, 1978.Google Scholar

[201] O. A., Ladyzhenskaya. The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1969.Google Scholar

[202] S., Lall, J. E., Marsden, and S., Glavaški. A subspace approach to balanced truncation for model reduction of nonlinear control systems. In t. J. Robust Nonlinear Control, 12:519–35, 2002.

[203] H., Lamb. Hydrodynamics. Dover, New York, sixth edition, 1945.Google Scholar

[204] L. D., Landau and E. M., Lifschitz. Fluid Mechanics. Pergamon Press, Oxford, UK, second edition, 1987.Google Scholar

[205] O., Lanford. Appendix to Lecture VII: computer pictures of the Lorenz attractor. In A., Chorin, J. E., Marsden, and S., Smale, editors, Turbulence Seminar, Berkeley, 1976/77, volume 615 of Springer Lecture Notes in Mathematics, pages 113–16. Springer-Verlag, New York, 1977.Google Scholar

[206] P., Laure and Y., Demay. Symbolic computation and the equation on the centermanifold: application to the Taylor-Couette problem. Comput. Fluids, 16:229–38, 1988.

[207] O., Lehmann, M., Luchtenburg, B. R., Noack, et al. Wake stabilization using POD Galerkin models with interpolated modes. In 44th IEEE Conference on Decision and Control and European Control Conference, pages 500–5, Dec. 2005.Google Scholar

[208] S., Leibovich. The form and dynamics of Langmuir circulations. Ann. Rev. Fluid Mech., 15:391–27, 1983.Google Scholar

[209] S., Leibovich. Structural genesis in wall bounded turbulent flows. In T. B., Gatski, S., Sarkar, and C. G., Speziale, editors, Studies in Turbulence, pages 387—411. Springer-Verlag, New York, 1992.Google Scholar

[210] A., Leonard and A., Wray. A new numerical method for simulation of three dimensional flow in a pipe. In E., Krause, editor, Proc. Int. Conf. on Numerical Methods in Fluid Dynamics, volume 170 of Lecture Notes in Physics, pages 335–2. Springer-Verlag, New York, 1982.Google Scholar

[211] H. W., Liepmann. Aspects of the turbulence problem, Part II. Z Angew. Math. Phys., 3:407–26, 1952.

[212] K. S., Lii, M., Rosenblatt, and C., Van Atta. Bispectral measurements in turbulence. J. Fluid Mech., 77:45–62, 1976.

[213] J. T., C. Liu. Contributions to the understanding of large-scale coherent structures in developing free turbulent shear flows. Advances in Applied Mechanics, 26:183–309, 1988.

[214] Z.-C., Liu, R. J., Adrian, and T. J., Hanratty. Reynolds number similarity of orthogonal decomposition of the outer layer of turbulent wall flow. Physics of Fluids, 6:2815–19, 1994.

[215] M., Loève. Functions aléatoire de second ordre. Comptes Rendus Acad. Set Paris, 220, 1945.Google Scholar

[216] E. N., Lorenz. Empirical orthogonal functions and statistical weather prediction. In Statistical Forecasting Project, Cambridge, MA, 1956, MIT Press.Google Scholar

[217] E. N., Lorenz. Deterministic nonperiodic flow. J. Atmos. Sci., 20:130–41, 1963.

[218] D. M., Luchtenburg, B. Günther, B. R., Noack, R., King, and G., Tadmor. A generalized mean-field model of the natural and high-frequency actuated flow around a high-lift configuration. J. FluidMech., 623:339–65, 2009.

[219] D. M., Luchtenburg, M., Schlegel, B. R., Noack, et al. Turbulence control based on reduced-order models and nonlinear control design. In R., King, editor, Active Flow Control II, vol 108 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, pages 341–56 Springer-Verlag, Berlin, 2010.Google Scholar

[220] J. L., Lumley. The structure of inhomogeneous turbulence. In A. M., Yaglom and V. I., Tatarski, editors, Atmospheric Turbulence and Wave Propagation, pages 166–78. Nauka, Moscow, 1967.Google Scholar

[221] J. L., Lumley. Stochastic Tools in Turbulence. Academic Press, New York, 1971.Google Scholar

[222] J. L., Lumley. Two-phase and non-Newtonian flows. In P., Bradshaw, editor, Turbulence. Topics in Applied Physics, Volume 12, pages 290–324, Springer-Verlag, New York, 1978.Google Scholar

[223] J. L., Lumley. Coherent structures in turbulence. In R. E., Meyer, editor, Transition and Turbulence, New York, 1981, Academic Press. Mathematics Research Center Symposia and Advanced Seminar Series.Google Scholar

[224] J. L., Lumley, editor. Whither Turbulence? Turbulence at the Crossroads, volume 357 of Lecture notes in Physics. Springer-Verlag, New York, 1990.

[225] J. L., Lumley. Early work on fluid mechanics in the IC engine. Ann. Rev. Fluid Mech., 33:319- 38,2001.

[226] J. L., Lumley and I., Kubo. Turbulent drag reduction by polymer additives: a survey. In B., Gampert, editor, The Influence of Polymer Additives on Velocity and Temperature Fields, pages 3–21. Springer-Verlag, New York, 1985.Google Scholar

[227] J. L., Lumley and H. A., Panofsky. The Structure of Atmospheric Turbulence. Interscience, New York, 1964.Google Scholar

[228] J. L., Lumley and A., Poje. Low-dimensional models for flows with density fluctuations. Physics of Fluids, 9(7):2023–31, 1997.

[229] A., Lundbladh, P., Schmidt, S., Berlin, and D., Henningson. Simulations of bypass transition for spatially evolving disturbances. In B., Cantwell, J., Jiménez, and S., Lekoudis, editors, Application of Direct and Large Eddy Simulation to Transition and Turbulence, pages 18.1–18.3. Fluid Dynamics Panel, NATO Advisory Group for Aerospace Research and Development, AGARDCP 551, 1994.Google Scholar

[230] X., Ma and G. E., Karniadakis. A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech., 458:181–90, 2002.

[231] Z., Ma, S., Ahuja, and C. W., Rowley. Reduced order models for control of fluids using the eigensystem realization algorithm. Theoret. Comput. Fluid Dynamics, 25(1):233–47, 2011.

[232] A., Mahalov and S., Leibovich. Multiple bifurcation of rotating pipe flow. Theoret. Comput. Fluid Dynamics, 3:61–77, 1992.

[233] S., Malo. Rigorous Computer Verification of Planar Vector Field Structure. PhD thesis, Cornell University, 1994.Google Scholar

[234] R., Mane. Ergodic Theory and Differentiable Dynamics. Springer-Verlag, New York, 1987.Google Scholar

[235] B., Marasli, R H., Champagne, and I., Wygnanski. Effect of traveling waves on the growth of a plane turbulent wake. J. Fluid Mech., 235:511–28, 1991.

[236] J. E., Marsden. Lectures on Mechanics, volume 174 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1992.Google Scholar

[237] T., Matsuoka and T. L., Ulrych. Phase estimation using the bispectrum. Proc. IEEE, 72:1403- 22, 1984.Google Scholar

[238] Y., Meyer. Wavelets: Algorithms and Applications. SIAM Publications, Philadelphia, PA, 1993.Google Scholar

[239] K., Mischaikow and M., Mrozek. Chaos in the Lorenz equations: a computer assisted proof. Bull. AMS (New Series), 32(1):66–72, 1995.

[240] H. K., Moffatt. Fixed points of turbulent dynamical systems and suppression of nonlinearity. In J. L., Lumley, editor, Whither Turbulence? Turbulence at the Crossroads, pages 250–7. Springer-Verlag, New York, 1990.Google Scholar

[241] P., Moin. Probing turbulence via large eddy simulation. AIAA paper 84-0174, 1984.

[242] P., Moin. Similarity of organized structures in turbulent shear flows. In S. J., Kline and N. H., Afgan, editors, Near-Wall Turbulence; 1988 Zoran Zaric Memorial Conference. Hemisphere Publishing, Washington, DC, 1990.Google Scholar

[243] P., Moin, R. J., Adrian, and J., Kim. Stochastic estimation of organized structures in turbulent channel flow. In Sixth Symposium on Turbulent Shear Flows, Toulouse, France, 1987. Ecole Nationale Superieure de l'Aeronautique et de l'Espace and ONERA Centre d'Etudes et de Recherches de Toulouse.Google Scholar

[244] P., Moin and R. D., Moser. Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech., 200:471–509, 1989.

[245] A. S., Monin and A. M., Yaglom. Statistical Fluid Mechanics: Mechanics of Turbulence, Volumes I and II. MIT Press, Cambridge, MA, 1971-1975.Google Scholar

[246] B. C., Moore. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control, 26(1):17–32, Feb. 1981.

[247] J., Moreau, M., Fogleman, G., Charnay, and J., Boree. Phase invariant proper orthogonal decomposition for the study of a compressed vortex. J. Thermal Sciences, 14(2): 108-13, 2005.Google Scholar

[248] M., Morzynski, W., Stankiewicz, B. R., Noack, R, Thiele, and G., Tadmor. Generalized mean-field model for flow control using continuous mode interpolation. 3rd AIAA Flow Control Conference, AIAA paper 2006-3488, June 2006.

[249] M., Morzynski, W., Stankiewicz, B. R., Noack, R Thiele, and G., Tadmor. Continuous mode interpolation for control-oriented models of fluid flow. In R., King, editor, Active Flow Control, vol 95 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, pages 260–78. Springer-Verlag, Berlin, 2007.Google Scholar

[250] R. D., Moser and M. M., Rogers. The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence. J. FluidMech., 247:275–320, 1993.

[251] T., Mullin. Disordered fluid motion in a small closed system. Physica D, 62:192–201, 1993.

[252] T., Mullin and A. G., Darbyshire. Intermittency in a rotating annular flow. Europhys. Lett., 9(7):669–73, 1989.

[253] T., Mullin, S. J., Taverner, and K. A., Cliffe. An experimental and numerical study of a codimension-2 bifurcation in a rotating annulus. Europhys. Lett., 8(3):251–6, 1989.

[254] M., Myers, P., Holmes, J., Elezgaray, and G., Berkooz. Wavelet projections of the Kuramoto-Sivashinsky equation I: Heteroclinic cycles and modulated traveling waves for short systems. Physica D, 86:396–427, 1995.

[255] M., Nagata. Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech., 217:519–27, 1990.

[256] R., Narasimha. The utility and drawbacks of traditional approaches. In J. L., Lumley, editor, Whither Turbulence? Turbulence at the Crossroads, pages 13–48. Springer-Verlag, New York, 1990.Google Scholar

[257] R., Narasimha and S. V., Kailas. Turbulent bursts in the atmosphere. Atmospheric Environment, 24A(7): 1635-45, 1990.Google Scholar

[258] S. E., Newhouse, D., Ruelle, and R Takens. Occurrence of strange axiom A attractors near quasiperiodic flows onTm, m > 3. Comm. Math. Phys., 64:35–10, 1978.Google Scholar

[259] B., Nicolaenko, B., Scheurer, and R., Temam. Some global dynamical properties of the Kuramoto-Sivashinsky equations: non-linear stability and attractors. Physica D, 16:155–83, 1985.

[260] B., Nicolaenko and Z., She. Temporal intermittency and turbulence production in the Kol-mogorov flow. In Topological Dynamics of Turbulence, pages 256–77. Cambridge University Press, Cambridge, UK, 1990.Google Scholar

[261] B., Nicolaenko and Z., She. Turbulent bursts, inertial sets and symmetry breaking homoclinic cycles in periodic Navier-Stokes flows. In G. R., Sell, C., Foias, and R., Temam, editors, Turbulence in Fluid Flows: a Dynamical Systems Approach, pages 123–36. Springer-Verlag, New York, 1993.Google Scholar

[262] B. R., Noack, K., Afanasiev, M., Morzynski, G., Tadmor, and F., Thiele. A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech., 497:335–63, 2003.

[263] B. R., Noack and H., Eckelmann. On chaos in wakes. Physica D, 56:151–64, 1992.

[264] B. R., Noack and H., Eckelmann. A global stability analysis of the steady and periodic cylinder wake. J. FluidMech., 270:297–330, 1994.

[265] B. R., Noack and H., Eckelmann. A low dimensional Galerkin method for the three-dimensional flow around a circular cylinder. Physics of Fluids, 6:124–43, 1994.

[266] B. R., Noack and H., Eckelmann. Theoretical investigation of the bifurcations and the turbulence attractor of the cylinder wake. Z Angew. Math. Mech., 74:T396-T397, 1994.Google Scholar

[267] B. R., Noack, M., Morzynski, and G., Tadmor (editors). Reduced-Order Modelling for Flow Control. Springer-Verlag, Berlin, 2010.Google Scholar

[268] B. R., Noack, P., Papas, and P. A., Monkewitz. The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. FluidMech., 523:283–316, 2005.

[269] B. R., Noack, G., Tadmor, and M., Morzynski. Actuation models and dissipative control in empirical Galerkin models of fluid flows. In American Control Conference, Boston, MA, June 30-July 2, 2004, pages 1–6, 2004.Google Scholar

[270] A., Novick-Cohen. Interfacial instabilities in directional solidification of dilute binary alloys: the Kuramoto-Sivashinsky equation. PhysicaD, 26:403–10, 1987.

[271] A., Novick-Cohen and G. I., Sivashinsky. On the solidification front of a dilute binary alloy: thermal diffusivity effects and breathing solutions. Physica D, 20:237–58, 1986.

[272] A. M., Obukhov. Statistical description of continuous fields. Trudy Geophys. Int. Aked. Nauk. SSSR, 24:3–42, 1954.

[273] D., Ornstein and B., Weiss. Statistical properties of chaotic systems. Bull. of the AMS (New Series), 24(1): 11-116, 1991.Google Scholar

[274] E., Ott, C., Grebogi, and J. A., Yorke. Controlling chaos. Phys. Rev. Lett., 64:1196–9, 1990.

[275] A., Papoulis. Probability, Random Variables, and Stochastic Processes. McGraw-Hill, New York, 1965.Google Scholar

[276] H., Park and L., Sirovich. Turbulent thermal convection in a finite domain, Part II. Numerical results. Physics of Fluids A, 2(9): 1659-68, 1990.Google Scholar

[277] M., Pastoor, B. R., Noack, R., King, and G., Tadmor. Spatiotemporal waveform observers and feedback in shear layer control. 44th AIAA Fluids Conference and Exhibit, AIAA paper 2006-1402, Jan. 2006.

[278] V., Perrier and C., Basdevant. Periodical wavelet analysis: A tool for inhomogeneous field investigation. theory and algorithms. Recherche Aerospatiale, 3:54–67, 1989.

[279] N., Platt, L., Sirovich, and N., Fitzmaurice. An investigation of chaotic Kolmogorov flows. Physics of Fluids A, 3(4):681–96, 1991.

[280] V. A., Pliss. A reduction principle in the theory of stability of motion. Izv. Akad. Nauk. SSSR. Math. Ser., 28:1297–324, 1964.

[281] B., Podvin, J., Gibson, G., Berkooz, and J. L., Lumley. Lagrangian and Eulerian views of the bursting period. Physics of Fluids, 9(2):433–7, 1997.

[282] H., Poincaré. Les Méthodes Nouvelles de la Mécanique Céleste, Tomes I-III. Gauthier-Villars, Paris, 1892, 1893, 1899. Reprinted 1987 by Libraire Albert Blanchard, Paris.Google Scholar

[283] A., Poje and J. L., Lumley. A model for large scale structures in turbulent shear flows. J. Fluid Mech., 285:349–69, 1995.

[284] S. B., Pope. PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci., 11:119–92, 1985.

[285] V. S., Pougachev. General theory of the correlations of random functions. Izv. Akad. Nauk. SSSR. Math. Ser., 17:401–2, 1953.

[286] R. W., Preisendorfer. Principal Component Analysis in Meteorology and Oceanography. Elsevier, Amsterdam, 1988.Google Scholar

[287] T. J., Price and T., Mullin. An experimental observation of a new type of intermittency. Physica A 48:29–52, 1991.

[288] M. K., Proctor and C., Jones. The interaction of two spatially resonant patterns in thermal convection 1: exact 1:2 resonance. J. FluidMech., 188:301–35, 1988.

[289] K., Promislow. Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations. Nonlinear Analysis: Theory, Methods and Applications, 16(11):959–80, 1991.

[290] M., Rajaee and S. K., F. Karlsson. Shear flow coherent structures via Karhunen-Loève expansion. Physics of Fluids A, 2:2249–51, 1990.

[291] M., Rajaee and S. K., F. Karlsson. On the Fourier space decomposition of free shear flow measurements and mode degeneration in the pairing process. Physics of Fluids A, 4:321–39, 1992.

[292] M., Rajaee, S. K., F. Karlsson, and L., Sirovich. Low-dimensional description of free-shear-flow coherent structures and their dynamical behaviour. J. Fluid Mech., 258:1–29, 1994.

[293] R. H., Rand. Computer Algebra in Applied Mathematics: An Introduction to MACSYMA, volume 94 of Research Notes in Mathematics. Pitman, Boston, MA, 1984.Google Scholar

[294] R. H., Rand and D., Armbruster. Perturbation Methods, Bifurcation Theory and Computer Algebra. Springer-Verlag, New York, 1987.Google Scholar

[295] S. C., Reddy and D. S., Henningson. Energy growth in viscous channel flows. J. FluidMech., 252:209–38, 1993.

[296] D., Rempfer. Low dimensional models of a flat-plate boundary layer. In R. M., C. So, C. G., Speciale, and B. E., Launder, editors, Near-Wall Turbulent Flows, pages 63–72. Elsevier, Amsterdam, 1993.Google Scholar

[297] D., Rempfer. On the structure of dynamical systems describing the evolution of coherent structures in a convective boundary layer. Physics of Fluids, 6(3):1402–1, 1994.

[298] D., Rempfer. On low-dimensional Galerkin models for fluid flow. Theoret. Comput. Fluid Dynamics, 14:75–88, 2000.

[299] D., Rempfer and H., Fasel. Evolution of coherent structures during transition in a flat-plate boundary layer. In Eighth Symposium on Turbulent Shear Flows, volume 1, pages 18.3.1–18.3.6, 1991.Google Scholar

[300] D., Rempfer and H., Fasel. The dynamics of coherent structures in a flat-plate boundary layer. Applied Scientific Research, 51:73–7, 1993.

[301] D., Rempfer and H., Fasel. Dynamics of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech., 275:257–83, 1994.

[302] D., Rempfer and H., Fasel. Evolution of three-dimensional coherent structures in a flat-plate boundary layer. J. Fluid Mech., 260:351–75, 1994.

[303] S. O., Rice. Mathematical analysis of random noise. Bell System TechnicalJ., 23 and 24:1–162, 1944. Reprinted in Selected Papers on Noise and Stochastic Processes, Wax, N., editor, Dover Publications Inc., New York, 1954.

[304] F., Riesz and B. S., Nagy. Functional Analysis. Ungar, New York, 1955.Google Scholar

[305] U., Rist and H., Fasel. Direct numerical simulation of controlled transition in a flat-plate boundary layer. J. Fluid Mech., 298:211–48, 1995.

[306] C., Robinson. Homoclinic bifurcation to a transitive attractor of Lorenz type. Nonlinearity, 2:495–518, 1989.

[307] S. K., Robinson. Coherent motions in the turbulent boundary layer. Ann. Rev. Fluid Mech., 23:601–39, 1991.

[308] J. D., Rodriguez and L., Sirovich. Low-dimensional dynamics for the complex Ginzburg- Landau equation. Physica D, 43:77–86, 1990.

[309] R. S., Rogallo and P., Moin. Numerical simulation of turbulent flows. Ann. Rev. Fluid Mech., 16:99–137, 1984.

[310] M. M., Rogers and P., Moin. The structure of the vorticity field in homogeneous turbulent flows. J. FluidMech., 176:33–66, 1987.

[311] M. M., Rogers and R. D., Moser. Spanwise scale selection in plane mixing layers. J. Fluid Mech., 247:321–37, 1993.

[312] V. A., Rokhlin. On the fundamental ideas of measure theory. AMS Transl. (1), 10:1–52, 1962.

[313] A., Rosenfeld and A. C., Kak. Digital Picture Processing. Academic Press, New York, 1982.Google Scholar

[314] C. W., Rowley. Modeling, Simulation, and Control of Cavity Flow Oscillations. PhD thesis, California Institute of Technology, 2002.Google Scholar

[315] C. W., Rowley. Model reduction for fluids using balanced proper orthogonal decomposition. International Journal of Bifurcation and Chaos, 15(3):997–1013, 2005.

[316] C. W., Rowley, T., Colonius, and R. M., Murray. Model reduction for compressible flows using POD and Galerkin projection. Physica D, 189(1-2):115-29, 2004.Google Scholar

[317] C. W., Rowley, I. G., Kevrekidis, J. E., Marsden, and K., Lust. Reduction and reconstruction for self-similar dynamical systems. Nonlinearity, 16:1257–75, 2003.

[318] C. W., Rowley and J. E., Marsden. Reconstruction equations and the Karhunen-Loève expansion for systems with symmetry. Physica D. Nonlinear Phenomena, 142:1–19, 2000.

[319] H. L., Royden. Real Analysis. Macmillan, London, 1963.Google Scholar

[320] D., Ruelle. Chaotic Evolution and Strange Attractors. Lezioni Lincee, Accademia Nazionale dei Lincei. Cambridge University Press, Cambridge, UK, 1989.Google Scholar

[321] D., Ruelle. Chance and Chaos. Princeton University Press, Princeton, NJ, 1991.Google Scholar

[322] D., Ruelle and R Takens. On the nature of turbulence. Comm. Math. Phys., 20:167–92, 1970. Addendum, 23, 343-4.

[323] S., Sanghi and N., Aubry. Mode interaction models for near-wall turbulence. J. Fluid Mech., 247:455–88, 1993.

[324] J. M., A. Scherpen. Balancing for nonlinear systems. Systems and Control Letters, 21(2):143- 53, 1993.

[325] P. J., Schmid and D. S., Henningson. Stability and Transition in Shear Flows. Springer-Verlag, New York, 2001.Google Scholar

[326] A., Schmiegel. Transition to Turbulence in Linearly Stable Shear Flows. PhD thesis, Universität Marburg, 1999.Google Scholar

[327] A., Schmiegel and B., Eckhardt. Fractal stability border in plane Couette flow. Phys. Rev. Lett., 79(26):5250–3, 1997.

[328] T.-H., Shih, J. L., Lumley, and J., Janica. Second order modeling of a variable density mixing layer. J. Fluid Mech., 180:93–116, 1987.

[329] S. G., Siegel, J., Seidel, C., Fagley, et al. Low dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition. J. Fluid Mech., 610:1–42, 2008.

[330] L., Sirovich. Turbulence and the dynamics of coherent structures, Parts I—III. Quarterly of Applied Math., XLV(3):561-82, 1987.Google Scholar

[331] L., Sirovich. Chaotic dynamics of coherent structures. Physica D, 37:126–43, 1989.

[332] L., Sirovich, K. S., Ball, and R. A., Handler. Propagating structures in wall-bounded turbulent flows. Theoret. Comput. Fluid Dynamics, 2:307–17, 1991.

[333] L., Sirovich, K. S., Ball, and L. R., Keefe. Plane waves and structures in turbulent channel flow. Physics of Fluids A, 2(12):2217–26, 1990.

[334] L., Sirovich and A. E., Deane. A computational study of Rayleigh-Bénard convection. Part II. Dimension considerations. J. FluidMech., 222:251–65, 1991.

[335] L., Sirovich, M., Kirby, and M., Winter. An eigenfunction approach to large scale transitional structures in jet flow. Physics of Fluids A, 2(2):127–36, 1990.

[336] L., Sirovich and B. W., Knight. The eigenfunction problem in higher dimensions: Asymptotic theory. Proc. Nat. Acad. Sci., 82:8275–8, 1985.

[337] L., Sirovich, M., Maxey, and H., Tarman. An eigenfunction analysis of turbulent thermal convection. In J.-C. André, J., Coustieux, E, Durst, et al., editors, Turbulent Shear Flows 6, pages 68–77. Springer-Verlag, New York, 1989.Google Scholar

[338] L., Sirovich and H., Park. Turbulent thermal convection in a finite domain, Part I. Theory. Physics of Fluids A, 2(9):1649–58, 1990.

[339] L., Sirovich and J. D., Rodriguez. Coherent structures and chaos: A model problem. Phys. Lett. A, 120(5):211–14, 1987.

[340] L., Sirovich and X., Zhou. Reply to “observations regarding ‘Coherence and chaos in a model of turbulent boundary layer’ by X., Zhou and L., Sirovich". Physics of Fluids A, 6:1579–82, 1994.

[341] G. I., Sivashinsky. Nonlinear analysis of hydrodynamic instability in laminar flames, Part I: Derivation of the basic equations. Acta Astronautica, 4:1176–206, 1977.

[342] S., Skogestad and I., Postlethwaite. Multivariable Feedback Control Analysis and Design. John Wiley and Sons, 2nd edition, 2005.Google Scholar

[343] S., Smale. Differentiable dynamical systems. Bull. AMS, 73:747–817, 1967.

[344] T., Smith and P., Holmes. Low dimensional models with varying parameters: A model problem and flow through a diffuser with variable angle. In J. L., Lumley, editor, Fluid Mechanics and the Environment: Dynamical Approaches, pages 315–36. Springer-Verlag, New York, 2001. Springer Lecture Notes in Physics 566.Google Scholar

[345] T. R., Smith, J., Moehlis, and P., Holmes. Dynamics of an 0:1:2 O(2)-equivarant system: Heteroclinic cycles and periodic orbits. Physica D, 211:347–76, 2005.

[346] T. R., Smith, J., Moehlis, and P., Holmes. Low-dimensional modelling of turbulence using the proper orthogonal decomposition: A tutorial. Nonlinear Dynamics, 41(1-3):275-307, 2005.Google Scholar

[347] T. R., Smith, J., Moehlis, and P., Holmes. Low-dimensional models for turbulent plane Couette flow in a minimal flow unit. J. Fluid Mech., 538:71–110, 2005.

[348] T. R., Smith, J., Moehlis, P., Holmes, and H., Faisst. Models for turbulent plane Couette flow using the proper orthogonal decomposition. Physics of Fluids, 14(7):2493–507, 2002.

[349] W. H., Snyder and J. L., Lumley. Some measurements of particle velocity autocorrelation functions in a turbulent flow. J. FluidMech., 48:41–71, 1971.

[350] C., Sparrow. The Lorenz Equations. Springer-Verlag, New York, 1982. [351] H. B., Squire. On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. Lond. A, 142:621–8, 1933.

[352] K. R., Sreenivasan, R., Narashima, and A., Prabhu. Zero-crossings in turbulent signals. J. Fluid Mech., 137:251–72, 1983.

[353] M. M., Stanišic. The Mathematical Theory of Turbulence. Springer-Verlag, New York, 1987.Google Scholar

[354] E., Stone. A Study of Low Dimensional Models for the Wall Region of a Turbulent Layer. PhD thesis, Cornell University, 1989.Google Scholar

[355] E., Stone and P., Holmes. Noise induced intermittency in a model of a turbulent boundary layer. Physica D, 37: 20–32, 1989.

[356] E., Stone and P., Holmes. Random perturbations of heteroclinic cycles. SIAM J. onAppl. Math., 50(3):726–43, 1990.

[357] E., Stone and P., Holmes. Unstable fixed points, heteroclinic cycles and exponential tails in turbulence production. Phys. Lett. A, 155:29–42, 1991.

[358] G., Strang. Linear Algebra and Its Applications. Academic Press, New York, 1980.Google Scholar

[359] D., Stretch, J., Kim, and R., Britter. A conceptual model for the structure of turbulent channel flow. In S., Robinson, editor, Notes for Boundary Layer Structure Workshop, Langley, VA, Aug. 1990. NASA.Google Scholar

[360] J. T., Stuart. On the non-linear mechanics of hydrodynamic stability. J. FluidMech., 4:1–21, 1958.

[361] H. L., Swinney and J. P., Gollub, editors. Hydrodynamic Instabilities and the Transition to Turbulence. Springer-Verlag, New York, second edition, 1985.Google Scholar

[362] G., Tadmor, O., Lehmann, B. R., Noack, and M., Morzynski. Mean field representation of the natural and actuated cylinder wake. Physics of Fluids, 22(3):034102, 2010.

[363] R., Tagg. The Couette-Taylor problem. Nonlinear Science Today, 4(3): 1–25, 1994.

[364] R., Tagg, D., Hirst, and H., Swinney. Critical dynamics near the spiral-Taylor vortex transition. Unpublished report, University of Texas, Austin, 1988.

[363] R., Tagg. The Couette–Taylor problem. Nonlinear Science Today, 4(3):1–25, 1994.

[365] F., Takens. Detecting strange attractors in turbulence. In D. A., Rand and L.-S., Young, editors, Dynamical Systems and Turbulence, Warwick 1980, volume 898 of Springer Lecture Notes in Mathematics, pages 366–81. Springer-Verlag, New York, 1981.Google Scholar

[366] J. A., Taylor and M. N., Glauzer. Towards practical flow sensing and control via POD and LSE based low-dimensional tools. A.S.M. E. J. Fluids Engineering, 126:337–45, 2004.

[367] R., Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics.Springer-Verlag, New York, 1988.Google Scholar

[368] H., Tennekes and J. L., Lumley. A First Course in Turbulence. MIT Press, Cambridge, MA, 1972.Google Scholar

[369] E. S., Titi. On approximate inertial manifolds to the Navier-Stokes equations. J. Math. Anal. Appl., 149:540–57, 1990.

[370] S., Toh. Statistical model with localized structures describing the spatio-temporal chaos of Kuramoto-Sivashinsky equation. J. Phys. Soc. Jap., 56(3):949–62, 1987.

[371] A. A., Townsend. The Structure of Turbulent Shear Flow. Cambridge University Press, Cambridge, UK, 1956.Google Scholar

[372] A. A., Townsend. Flow patterns of large eddies in a wake and in a boundary layer. J. Fluid Mech., 95:515–37, 1979.

[373] L. N., Trefethen and D. I., Bau. Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997.Google Scholar

[374] L., Ukeiley. Dynamics of Large Scale Structures in a Plane Turbulent Mixing Layer. PhD thesis, Clarkson University, 1995.Google Scholar

[375] B. van der, Pol. Forced oscillations in a circuit with nonlinear resistance (receptance with reactive diode). London, Edinburgh and Dublin Phil. Mag., 3:65–80, 1927.CrossRef

[376] M. I., Vishik. Asymptotic Behaviour of Solutions of Evolutionary Equations. Lezioni Lincee, Accademia Nazionale dei Lincei. Cambridge University Press, Cambridge, UK, 1992.Google Scholar

[377] F., Waleffe. Hydrodynamic stability and turbulence: Beyond transients to a self-sustaining process. Stud. Appl. Math., 95:319–43, 1995.

[378] F., Waleffe. Transition in shear flows. Nonlinear normality versus non-normal linearity. Physics of Fluids, 7(12):3060–6, 1995.

[379] F., Waleffe. On a self-sustaining process in shear flows. Physics of Fluids, 9:883–900, 1997.

[380] F., Waleffe. Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett., 81: 4140-3, 1998.Google Scholar

[381] F., Waleffe. Exact coherent structures in channel flow. J. Fluid Mech., 435:93–102, 2001.

[382] F., Waleffe, J., Kim, and J. M., Hamilton. On the origin of streaks in turbulent shear flows. In F., Durst, R., Friedrich, B. E., Launder, et al., editors, Turbulent Shear Flows 8, pages 37–49. Springer-Verlag, New York, 1991.Google Scholar

[383] Y., Wang and H. H., Bau. Period doubling and chaos in a thermal convection loop with time periodic wall temperature variation. In G., Hetzroni, editor, Proc. 9th International Heat Transfer Conf. VolII, pages 357–62, 1990.Google Scholar

[384] Y., Wang, J., Singer, and H. H., Bau. Controlling chaos in a thermal convection loop. J. Fluid Mech., 237:479–98, 1992.

[385] J., Weller, E., Lombardi, and A., Iollo. Robust model identification of actuated vortex wakes. Physica D, 238:416–27, 2009.

[386] G. B., Whitham. Linear and Nonlinear Waves. Wiley, New York, 1974.Google Scholar

[387] P. J., Widmann, M., Gorman, and K. A., Robbins. Nonlinear dynamics of a convection loop II: chaos in laminar and turbulent flows. Physica D, 36:157–66, 1989.

[388] M., Winter, T. J., Barber, R. M., Everson, and L., Sirovich. Eigenfunction analysis of turbulent mixing phenomena. AIAA Journal, 30(7):1681–8, 1992.

[389] R. W., Wittenberg and P., Holmes. Scale and space localisation in the Kuramoto-Sivashinsky equation. Chaos, 9(2):452–65, 1999.

[390] R. W., Wittenberg and P., Holmes. Spatially localized models of extended systems. Nonlinear Dynamics, 25:111–32, 2001.

[391] M., Yokokawa, K., Itakura, A., Uno, T., Ishihara, and Y Kaneda. 16.4 Tflops direct numerical simulation of turbulence by a Fourier spectral method on the Earth simulator. In Proceedings of the ACM/IEEE Conference on Supercomputing, 2002.Google Scholar

[392] X., Zheng and M. N., Glauser. A low dimensional description of the axisymmetric jet mixing layer. ASME Computers in Engineering, 2:121–7, 1990.

[393] K., Zhou, G., Salomon, and E., Wu. Balanced realization and model reduction for unstable systems. International Journal of Robust and Nonlinear Control, 9(3):183–98, 1999.

[394] X., Zhou and L., Sirovich. Coherence and chaos in a model of turbulent boundary layer. Physics of Fluids A, 4:2855–74, 1992.

[395] Y., Zhou and G., Vahala. Local interaction in renormalization methods for Navier-Stokes turbulence. Phys. Rev. A, 46:1136–9, 1992.

[396] Y., Zhou and G., Vahala. Reformulation of recursive renormalization group based subgrid modeling of turbulence. Phys. Rev. E, 47:2503–19, 1993.