On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain

SIAM Journal on Applied Dynamical Systems

Volumn 9, Issue 1, 2010, Pages 1-33

  Cvitanović, Predrag ^a   Davidchack, Ruslan L ^b   Siminos, Evangelos ^a  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF LEICESTER   (United Kingdom)

Author keywords

Chaos; Continuous symmetry; Kuramoto Sivashinsky equation; Relative periodic orbits; Turbulence

Indexed keywords

CHAOTIC DYNAMICS; CONTINUOUS SYMMETRY; DECOMPOSITION OF STATE SPACE; DISCRETE SYMMETRY; DYNAMICAL STATE; ENERGY-TRANSFER RATE; FLOWTHROUGH; HETEROCLINIC CONNECTION; HIGH-DIMENSIONAL; INVARIANT SET; INVARIANT SUBSPACE; KURAMOTO-SIVASHINSKY; KURAMOTO-SIVASHINSKY EQUATION; KURAMOTO-SIVASHINSKY EQUATIONS; PERIODIC DOMAINS; PERIODIC ORBITS; RELATIVE EQUILIBRIA; SPATIAL TRANSLATION; STATE SPACE; SYMMETRY INVARIANT; TRAVELING WAVE;

CHAOS THEORY; ENERGY TRANSFER; SPUTTERING; TIME VARYING SYSTEMS; TURBULENCE;

ORBITS;

EID: 74049146819     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/070705623     Document Type: Article

References (50)

1. D. ARMBRUSTER, J. GUCKENHEIMER, AND P. HOLMES, Heteroclinic cycles and modulated travelling waves in systems with O (2) symmetry, Phys. D, 29 (1988), pp. 257-282.
2. D. ARMBRUSTER, J. GUCKENHEIMER, AND P. HOLMES, Kuramoto-Sivashinsky dynamics on the centerunstable manifold, SIAM J. Appl. Math., 49 (1989), pp. 676-691.
3. 2 bounds for the Kuramoto-Sivashinsky equation, Nonlinearity, 19 (2006), pp. 2023-2039. arXiv:math/0508481
4. H. S. BROWN AND I. G. KEVREKIDIS, Modulated traveling waves for the Kuramoto-Sivashinsky equation, in Pattern Formation: Symmetry Methods an d Applications, D. Benest and C. Froeschlé, eds., Fields Inst. Commun. 5, AMS, Providence, RI, 1996, pp. 45-66.
5. A. CHENCINER, Three Body Problem, Web article, 2007; online at http://scholarpedia.org/article/Three-body-problem.
6. F. CHRISTIANSEN, P. CVITANOVIĆ, AND V. PUTKARADZE, Spatiotemporal chaos in terms of unstable recurrent patterns, Nonlinearity, 10 (1997), pp. 55-70.
7. S. M. COX AND P. C. MATTHEWS, Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), pp. 430-455.
8. J. J. CROFTS, Efficient Method for Detection of Periodic Orbits in Chaotic Maps and Flows, Ph. D. thesis, Department of Mathematics, University of Leicester, Leicester, UK, 2007. arXiv:nlin. CD/0706.1940
9. J. J. CROFTS AND R. L. DAVIDCHASK, On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows, Chaos, 19 (2009), paper 033138.
10. M. C. CROSS AND P. C. HOHENBERG, Pattern formation outside of equilibrium, Rev. Mod. Phys, 65 (1993), pp. 851-1112.
11. P. CVITANOVIĆ, Continuous symmetry reduced trace formulas, in Chaos: Classical and Quantum, P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, eds., Web book, Niels Bohr Institute, Copenhagen, Denmark, http://ChaosBook.org/~predrag/papers/trace.pdf, 2007.
12. P. CVITANOVIĆ, R. ARTUSO, R. MAINIERI, G. TANNER, AND G. VATTAY, EDS., Chaos: Classical and Quantum, Niels Bohr Institute, Copenhagen, 2008; http://ChaosBook.org/version12.
13. P. CVITANOVIĆ AND B. ECKHARDT, Symmetry decomposition of chaotic dynamics, Nonlinearity, 6 (1993), pp. 277-311. arXiv:chao-dyn/9303016
14. C. FOIAS, B. NICOLAENKO, G. R. SELL, AND R. TEMAM, Inertial manifold for the Kuramoto-Sivashinsky equation, C. R. Acad. Sci. Paris Ser. I, 301 (1985), pp. 285-288.
15. M. FRIGO AND S. G. JOHNSON, The design and implementation of FFTW3, Proc. IEEE, 93 (2005), pp. 216-231.
16. U. FRISCH, Turbulence, Cambridge University Press, Cambridge, UK, 1996.
17. U. FRISCH, Z. S. SHE, AND O. THUAL, Viscoelastic behavior of cellular solutions to the Kuramoto-Sivashinsky model, J. Fluid Mech., 168 (1986), pp. 221-240.
18. L. GIACOMLLI AND F. OTTO, New bounds for the Kuramoto-Sivashinsky equation, Comm. Pure Appl. Math., 58 (2005), pp. 297-318.
19. J. F. GIBSON, Movies of Plane Couette, Technical report, Georgia Institute of Technology, Atlanta, GA, 2008, online at http://ChaosBook.org/tutorials.
20. J. F. GIBSON, J. HALCROW, AND P. CVITANOVIĆ, Visualizing the geometry of state-space in plane Couette flow, J. Fluid Mech., 611 (2008), pp. 107-130. arXiv:0705.3957
21. F. GINELLI, P. POGGI, A. TURCHI, H. CHATÉ, R. LIVI, AND A. POLITI, Characterizing dynamics with covariant Lyapunov vectors, Phys. Rev. Lett., 99 (2007), paper 130601. arXiv:0706.0510
22. M. GOLUBITSKY AND I. STEWART, The Symmetry Perspective, Birkhäuser Boston, Cambridge, MA, 2002.
23. J. M. GREENE AND J. S. KIM, The steady states of the Kuramoto-Sivashinsky equation, Phys. D, 33 (1988), pp. 99-120.
24. J. HAMILTON, J. KIM, AND F. WALEFFE, Regeneration mechanisms of near-wall turbulence structures, J. Fluid Mech., 287 (1995), pp. 317-348.
25. B. HOF, C. W. H. VAN DOORNE, J. WESTERWEEL, F. T. M. NIEUWSTADT, H. FAISST, B. ECKHARDT, H. WEDIN, R. R. KERSWELL, AND F. WALEFFE, Experimental observation of nonlinear traveling waves in turbulent pipe flow, Science, 305 (2004), pp. 1594-1598.
26. P. HOLMES, J. L. LUMLEY, AND G. BERKOOZ, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 1996.
27. E. HOPF, A mathematical example displaying features of turbulence, Comm. Appl. Math., 1 (1948), pp. 303-322.
28. J. M. HYMAN, B. NICOLAENKO, AND S. ZALESKI, Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces, Phys. D, 23 (1986), pp. 265-292.
29. M. E. JOHNSON, M. S. JOLLY, AND I. G. KEVREKIDIS, The Oseberg transition: Visualization of global bifurcations for the Kuramoto-Sivashinsky equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 1-18.
30. M. JOLLY, R. ROSA, AND R. TEMAM, Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation, Adv. Differential Equat ions, 5 (2000), pp. 31-66.
31. A.-K. KASSAM AND L. N. TREFETHEN, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26 (2005), pp. 1214-1233.
32. G. KAWAHARA AND S. KIDA, Periodic motion embedded in plane Couette turbulence: Regeneration cycle and burst, J. Fluid Mech., 449 (2001), pp. 291-300.
33. I. G. KEVREKIDIS, B. NICOLAENKO, AND J. C. SCOVEL, Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Math., 50 (1990), pp. 760-790.
34. B. KRAUSKOPF, H. OSINGA, E. DOEDEL, M. HENDERSON, J. GUCKENHEIMER, A. VLADIMIRSKY, M. DELLNITZ, AND O. JUNGE, A survey of methods for computing (un) stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), pp. 763-791.
35. M. KRUPA, Bifurcations of relative equilibria, SIAM J. Math. Anal., 21 (1990), pp. 1453-1486.
36. M. KRUPA, Robust heteroclinic cycles, J. Nonlinear Sci., 7 (1997), pp. 129-176.
37. Y. KURAMOTO AND T. TSUZUKI, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progr. Theoret. Phys., 55 (1976), p. 365.
38. Y. LAN, Dynamical Systems Approach to One-Dimensional Spatiotemporal Chaos-A Cyclist's View, Ph. D. thesis, School of Physics, Georgia Institute of Technology, Atlanta, GA, 2004; available online at http://etd.gatech.edu/theses/available/etd-10282004-154606/.
39. Y. LAN AND P. CVITANOVIĆ, Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics, Phys. Rev. E, 78 (2008), paper 026208. arXiv.org:0804.2474
40. R. E. LAQUEY, S. M. MAHAJAN, P. H. RUTHERFORD, AND W. M. TANG, Nonlinear saturation of the trapped-ion mode, Phys. Rev. Lett., 34 (1974), pp. 391-394.
41. V. LÓPEZ, P. BOYLAND, M. T. HEATH, AND R. D. MOSER, Relative periodic solutions of the complex Ginzburg-Landau equation, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 1042-1075.
42. P. MANNEVILLE, Dissipative Structures and Weak Turbulence, Academic Press, Boston, 1990.
43. D. MICHELSON, Steady solutions of the Kuramoto-Sivashinsky equation, Phys. D, 19 (1986), pp. 89-111.
44. J. J. MORÉ, B. S. GARBOW, AND K. E. HILLSTROM, User Guide for MINPACK-1, report ANL-80-74, Argonne National Laboratory, Argonne, IL, 1980.
45. E. SIMINOS, Recurrent Spatio-Temporal Structures in Presence of Continuous Symmetries, Ph. D. thesis, School of Physics, Georgia Institute of Technology, Atlanta, GA, 2009; available online at http://ChaosBook.org/projects/Siminos/thesis.pdf.
46. E. SIMINOS, P. CVITANOVIĆ, AND R. L. DAVIDCHACK, Recurrent spatio-temporal structures of translationally invariant Kuramoto-Sivashinsky flow, manuscript, 2009.
47. G. I. SIVASHINSKY, Nonlinear analysis of hydrodynamical instability in laminar flames-I. Derivation of basic equations, Acta Astronaut., 4 (1977), pp. 1177-1206.
48. V. SZEBEHELY, Theory of Orbits, Academic Press, New York, 1967.
49. D. VISWANATH, Recurrent motions within plane Couette turbulence, J. Fluid Mech., 580 (2007), pp. 339-358. arXiv:physics/0604062
50. H.-L. YANG, K. A. TAKEUCHI, F. GINELLI, H. Chaté, AND G. RADONS, Hyperbolicity and the effective dimension of spatially-extended dissipative systems, Phys. Rev. Lett., 102 (2009), paper 074102. arXiv: 0807.5073