Auerbach, D., Cvitanović, P., Eckmann, J.-P., Gunaratne, G. & Procaccia, I.
1987
Exploring chaotic motion through periodic orbits. Phys. Rev. Lett.
58, 23.
Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B.
2011
The onset of turbulence in pipe flow. Science
333, 192–196.
Avila, M., Willis, A. P. & Hof, B.
2010
On the transient nature of localized pipe flow turbulence. J. Fluid Mech.
646, 127–136.
Beyn, W.-J. & Thümmler, V.
2004
Freezing solutions of equivariant evolution equations. SIAM J. Appl. Dyn. Syst.
3, 85–116.
Cartan, É.
1935
La méthode du repère mobile, la théorie des groupes continus, et les espaces généralisés. In Exposés de Géométrie, vol. 5. Hermann.
Christiansen, F., Cvitanović, P. & Putkaradze, V.
1997
Spatio-temporal chaos in terms of unstable recurrent patterns. Nonlinearity
10, 55–70.
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G. & Vattay, G.
2012
Chaos: Classical and Quantum. Niels Bohr Inst., www.chaosbook.org.
Cvitanović, P., Borrero-Echeverry, D., Carroll, K., Robbins, B. & Siminos, E.
2012
Cartography of high-dimensional flows: a visual guide to sections and slices. Chaos
22, 047506.
Cvitanović, P., Davidchack, R. L. & Siminos, E.
2009
On the state space geometry of the Kuramoto–Sivashinsky flow in a periodic domain. SIAM J. Appl. Dyn. Syst.
9, 1–33.
Cvitanović, P. & Gibson, J. F.
2010
Geometry of turbulence in wall-bounded shear flows: periodic orbits. Phys. Scr. T
142, 014007.
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R.
2008a
Relative periodic orbits in transitional pipe flow. Phys. Fluids
20, 114102.
Duguet, Y., Willis, A. P. & Kerswell, R. R.
2008b
Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech.
613, 255–274.
Faisst, H. & Eckhardt, B.
2003
Traveling waves in pipe flow. Phys. Rev. Lett.
91, 224502.
Fels, M. & Olver, P. J.
1998
Moving coframes: I. A practical algorithm. Acta Appl. Math.
51, 161–213.
Fels, M. & Olver, P. J.
1999
Moving coframes: II. Regularization and theoretical foundations. Acta Appl. Math.
55, 127–208.
Frisch, U.
1996
Turbulence. Cambridge University Press.
Froehlich, S. & Cvitanović, P.
2011
Reduction of continuous symmetries of chaotic flows by the method of slices. Commun. Nonlinear Sci. Numer. Simul.
17, 2074–2084.
Gibson, J. F., Halcrow, J. & Cvitanović, P.
2008
Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech.
611, 107–130.
Gibson, J. F., Halcrow, J. & Cvitanović, P.
2009
Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech.
638, 1–24.
Greene, J. M. & Kim, J.-S.
1988
The steady states of the Kuramoto–Sivashinsky equation. Physica D
33, 99–120.
Halcrow, J., Gibson, J. F., Cvitanović, P. & Viswanath, D.
2009
Heteroclinic connections in plane Couette flow. J. Fluid Mech.
621, 365–376.
Hamilton, J. M., Kim, J. & Waleffe, F.
1995
Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech.
287, 317–348.
Hof, B., De Lozar, A., Kuik, D. J. & Westerweel, J.
2008
Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow. Phys. Rev. Lett.
101, 214501.
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F.
2004
Experimental observation of nonlinear travelling waves in turbulent pipe flow. Science
305, 1594–1598.
Hopf, E.
1948
A mathematical example displaying features of turbulence. Comm. Pure Appl. Maths
1, 303–322.
Huygens, C.
1967
L’Horloge à pendule de 1673. Swets & Zeitlinger.
Jiménez, J. & Moin, P.
1991
The minimal flow unit in near-wall turbulence. J. Fluid Mech.
225, 213–240.
Kawahara, G. & Kida, S.
2001
Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech.
449, 291–300.
Kawahara, G., Uhlmann, M. & van Veen, L.
2012
The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech
44, 203–225.
Kerswell, R. R. & Tutty, O. R.
2007
Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech.
584, 69–102.
Kim, K. C. & Adrian, R. J.
1999
Very large-scale motion in the outer layer. Phys. Fluids
11, 417–422.
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W.
1967
The structure of turbulent boundary layers. J. Fluid Mech.
30, 741–773.
Kreilos, T. & Eckhardt, B.
2012
Periodic orbits near onset of chaos in plane Couette flow. Chaos
22, 047505.
Lan, Y. & Cvitanović, P.
2008
Unstable recurrent patterns in Kuramoto–Sivashinsky dynamics. Phys. Rev. E
78, 026208.
Lombardi, M., Caulfield, C. P., Cossu, C., Pesci, A. I. & Goldstein, R. E.
2011
Growth and instability of a laminar plume in a strongly stratified environment. J. Fluid Mech.
671, 184–206.
Mellibovsky, F. & Eckhardt, B.
2012
From travelling waves to mild chaos: A supercritical bifurcation cascade in pipe flow. J. Fluid Mech.
709, 149–190.
Mellibovsky, F. & Eckhardt, B.
2011
Takens–Bogdanov bifurcation of travelling-wave solutions in pipe flow. J. Fluid Mech.
670, 96–129.
Mullin, T. & Kerswell, R. R.
2005
Non-uniqueness of Solutions to the Navier–Stokes Equations and their Connection with Laminar–Turbulent Transition. Kluwer.
Olver, P. J.
1999
Classical Invariant Theory. Cambridge University Press.
Poincaré, H.
1896
Sur les solutions périodiques et le principe de moindre action. C. R. Acad. Sci. Paris
123, 915–918.
Pringle, C. C. T., Duguet, Y. & Kerswell, R. R.
2009
Highly symmetric travelling waves in pipe flow. Phil. Trans. R. Soc. A
367, 457–472.
Pringle, C. C. T. & Kerswell, R. R.
2007
Asymmetric, helical, and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett.
99, 074502.
Rand, D.
1982
Dynamics and symmetry – predictions for modulated waves in rotating fluids. Arch. Rational Mech. Anal.
79, 1–3.
Recke, L., Samoilenko, A., Tkachenko, V. & Yanchuk, S.
2011 Frequency locking by external forcing in systems with rotational symmetry. Arxiv preprint. arXiv:1108.5990.
Rowley, C. W. & Marsden, J. E.
2000
Reconstruction equations and the Karhunen–Loéve expansion for systems with symmetry. Physica D
142, 1–19.
Schneider, T. M., Eckhardt, B. & Vollmer, J.
2007
Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E
75, 066313.
Schneider, T. M., Gibson, J. F. & Burke, J.
2010
Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett.
104, 104501.
Siminos, E. & Cvitanović, P.
2011
Continuous symmetry reduction and return maps for high-dimensional flows. Physica D
240, 187–198.
Tempelmann, D., Hanifi, A. & Henningson, D. S.
2010
Spatial optimal growth in three-dimensional boundary layers. J. Fluid Mech.
646, 5–37.
Viswanath, D.
2007
Recurrent motions within plane Couette turbulence. J. Fluid Mech.
580, 339–358.
Wedin, H. & Kerswell, R. R.
2004
Exact coherent structures in pipe flow: traveling wave solutions. J. Fluid Mech.
508, 333–371.
Willis, A. P. & Kerswell, R. R.
2008
Coherent structures in localised and global pipe turbulence. Phys. Rev. Lett.
100, 124501.
