Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 636–655 | Cite as

Unstable Manifolds of Relative Periodic Orbits in the Symmetry-Reduced State Space of the Kuramoto–Sivashinsky System

Article

Abstract

Systems such as fluid flows in channels and pipes or the complex Ginzburg–Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto–Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.

Keywords

Kuramoto–Sivashinsky equation Equivariant systems Relative periodic orbits Unstable manifolds Chaos Symmetries 

Notes

Acknowledgements

This work was supported by the family of late G. Robinson, Jr. and NSF Grant DMS-1211827. We are grateful to Xiong Ding, Evangelos Siminos, Simon Berman, and Mohammad Farazmand for many fruitful discussions.

References

  1. 1.
    Armbruster, D., Guckenheimer, J., Holmes, P.: Kuramoto-Sivashinsky dynamics on the center-unstable manifold. SIAM J. Appl. Math. 49, 676–691 (1989)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, Berlin (1982)Google Scholar
  3. 3.
    Artuso, R., Aurell, E., Cvitanović, P.: Recycling of strange sets: I. Cycle expansions. Nonlinearity 3, 325–359 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Artuso, R., Aurell, E., Cvitanović, P.: Recycling of strange sets: II. Applications. Nonlinearity 3, 361–386 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Avila, M., Mellibovsky, F., Roland, N., Hof, B.: Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110, 224502 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Blackburn, H.M., Marques, F., Lopez, J.M.: Symmetry breaking of two-dimensional time-periodic wakes. J. Fluid Mech. 522, 395–411 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer, Berlin (1975)CrossRefMATHGoogle Scholar
  8. 8.
    Budanur, N.B.: Exact coherent structures in spatiotemporal chaos: from qualitative description to quantitative predictions. PhD thesis, School of Physics, Georgia Inst. of Technology, Atlanta (2015)Google Scholar
  9. 9.
    Budanur, N.B., Hof, B.: State space geometry of the laminar-turbulent boundary in pipe flow, in preparation (2017)Google Scholar
  10. 10.
    Budanur, N.B., Borrero-Echeverry, D., Cvitanović, P.: Periodic orbit analysis of a system with continuous symmetry: a tutorial. Chaos 25, 073112 (2015)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Budanur, N.B., Cvitanović, P., Davidchack, R.L., Siminos, E.: Reduction of the SO(2) symmetry for spatially extended dynamical systems. Phys. Rev. Lett. 114, 084102 (2015)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Cartan, E.: La méthode du repère mobile, la théorie des groupes continus, et les espaces généralisés, Vol. 5, Exposés de Géométrie. Hermann, Paris (1935)MATHGoogle Scholar
  13. 13.
    Chossat, P., Lauterbach, R.: Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific, Singapore (2000)CrossRefMATHGoogle Scholar
  14. 14.
    Christiansen, F., Cvitanović, P., Putkaradze, V.: Spatiotemporal chaos in terms of unstable recurrent patterns. Nonlinearity 10, 55–70 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Crawford, J.D., Knobloch, E.: Symmetry and symmetry-breaking bifurcations in uid dynamics. Ann. Rev. Fluid Mech. 23, 341–387 (1991)ADSCrossRefMATHGoogle Scholar
  16. 16.
    Cvitanović, P.: Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett. 61, 2729–2732 (1988)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Cvitanović, P., Davidchack, R.L., Siminos, E.: On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain. SIAM J. Appl. Dyn. Syst. 9, 1–33 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Cvitanović, P., Borrero-Echeverry, D., Carroll, K., Robbins, B., Siminos, E.: Cartography of high-dimensional flows: a visual guide to sections and slices. Chaos 22, 047506 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G.: Chaos: Classical and Quantum. Niels Bohr Institute, Copenhagen (2016)Google Scholar
  20. 20.
    D’Humieres, D., Beasley, M.R., Huberman, B.A., Libchaber, A.: Chaotic states and routes to chaos in the forced pendulum. Phys. Rev. A 26, 3483–3496 (1982)ADSCrossRefGoogle Scholar
  21. 21.
    Duguet, Y., Willis, A.P., Kerswell, R.R.: Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255–274 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Farazmand, M.: An adjoint-based approach for finding invariant solutions of Navier-Stokes equations. J. Fluid Mech. 795, 278–312 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Field, M.J.: Equivariant dynamical systems. Trans. Am. Math. Soc. 259, 185–205 (1980)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Gatermann, K.: Computer Algebra Methods for Equivariant Dynamical Systems. Springer, New York (2000)CrossRefMATHGoogle Scholar
  26. 26.
    Gibson, J.F.: Channel flow: a spectral Navier-Stokes simulator in C++, technical report. University of New Hampshire. www.Channelflow.org (2013)
  27. 27.
    Gibson, J.F., Halcrow, J., Cvitanović, P.: Visualizing the geometry of state-space in plane Couette flow. J. Fluid Mech. 611, 107–130 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Gilmore, R., Letellier, C.: The Symmetry of Chaos. Oxford University Press, Oxford (2007)MATHGoogle Scholar
  29. 29.
    Greene, J.M., Kim, J.-S.: The steady states of the Kuramoto-Sivashinsky equation. Physica D 33, 99–120 (1988)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Gutzwiller, M.C.: Periodic orbits and classical quantization conditions. J. Math. Phys. 12, 343–358 (1971)ADSCrossRefGoogle Scholar
  31. 31.
    Hilbert, D.: Über die vollen Invariantensysteme. Math. Ann. 42, 313–373 (1893)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Hindmarsh, A.C.: ODEPACK, a systematized collection of ODE solvers. In: Stepleman, R.S. (ed.) Scientific Computing, vol. 1, pp. 55–64. North-Holland, Amsterdam (1983)Google Scholar
  33. 33.
    Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996)CrossRefMATHGoogle Scholar
  34. 34.
    Hopf, E.: Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 94, 3–22 (1942)Google Scholar
  35. 35.
    Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: open source scientific tools for Python (2001)Google Scholar
  36. 36.
    Kadanoff, L., Tang, C.: Escape rate from strange repellers. Proc. Natl. Acad. Sci. USA 81, 1276–1279 (1984)ADSCrossRefMATHGoogle Scholar
  37. 37.
    Kevrekidis, I.G., Nicolaenko, B., Scovel, J.C.: Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation. SIAM J. Appl. Math. 50, 760–790 (1990)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Knobloch, E., Weiss, N.: Bifurcations in a model of double-diffusive convection. Phys. Lett. A 85, 127–130 (1981)ADSCrossRefGoogle Scholar
  39. 39.
    Krupa, M.: Bifurcations of relative equilibria. SIAM J. Math. Anal. 21, 1453–1486 (1990)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Kuramoto, Y., Tsuzuki, T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55, 356–369 (1976)ADSCrossRefGoogle Scholar
  41. 41.
    Lan, Y., Cvitanović, P.: Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics. Phys. Rev. E 78, 026208 (2008)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Lan, Y., Chandre, C., Cvitanović, P.: Variational method for locating invariant tori. Phys. Rev. E 74, 046206 (2006)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Mainieri, R., Cvitanović, P. (ed.): A brief history of chaos. In: Chaos: Classical and Quantum. Niels Bohr Institute, Copenhagen (2016)Google Scholar
  44. 44.
    Marques, F., Lopez, J.M., Blackburn, H.M.: Bifurcations in systems with Z2 spatio-temporal and O(2) spatial symmetry. Physica D 189, 247–276 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Miranda, R., Stone, E.: The proto-Lorenz system. Phys. Lett. A 178, 105–113 (1993)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Neimark, J.: On some cases of periodic motions depending on parameters. Dokl. Akad. Nauk SSSR 129, 736–739 (1959). in RussianMathSciNetGoogle Scholar
  47. 47.
    Noether, E.: Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann. 77, 89–92 (1915)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Novak, S., Frehlich, R.G.: Transition to chaos in the Duffing oscillator. Phys. Rev. A 26, 3660–3663 (1982)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Platt, N., Sirovich, L., Fitzmaurice, N.: An investigation of chaotic Kolmogorov flows. Phys. Fluids A 3, 681–696 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Rempel, E.L., Chian, A.C.: Intermittency induced by attractor-merging crisis in the Kuramoto-Sivashinsky equation. Phys. Rev. E 71, 016203 (2005)ADSCrossRefGoogle Scholar
  51. 51.
    Rempel, E.L., Chian, A.C., Macau, E.E., Rosa, R.R.: Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto-Sivashinsky equation. Chaos 14, 545–556 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Rempel, E.L., Chian, A.C., Miranda, R.A.: Chaotic saddles at the onset of intermittent spatiotemporal chaos. Phys. Rev. E 76, 056217 (2007)ADSCrossRefGoogle Scholar
  53. 53.
    Ruelle, D.: Generalized zeta-functions for Axiom A basic sets. Bull. Am. Math. Soc. 82, 153–156 (1976)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Ruelle, D.: Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34, 231–242 (1976)ADSMathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Sacker, R.J.: A new approach to the perturbation theory of invariant surfaces. Commun. Pure Appl. Math. 18, 717–732 (1965)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Schneider, T.M., Eckhardt, B., Yorke, J.: Turbulence, transition, and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502 (2007)ADSCrossRefGoogle Scholar
  57. 57.
    Siminos, E.: Recurrent spatio-temporal structures in presence of continuous symmetries. PhD thesis, School of Physics, Georgia Institute of Technology, Atlanta (2009)Google Scholar
  58. 58.
    Siminos, E., Cvitanović, P.: Continuous symmetry reduction and return maps for high-dimensional flows. Physica D 240, 187–198 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Sivashinsky, G.I.: Nonlinear analysis of hydrodynamical instability in laminar ames—I. Derivation of basic equations. Acta Astronaut. 4, 1177–1206 (1977)ADSMathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Skufca, J.D., Yorke, J.A., Eckhardt, B.: Edge of Chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101 (2006)ADSCrossRefGoogle Scholar
  61. 61.
    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Swift, J.W., Wiesenfeld, K.: Suppression of period doubling in symmetric systems. Phys. Rev. Lett. 52, 705–708 (1984)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    Toh, S., Itano, T.: A periodic-like solution in channel flow. J. Fluid Mech. 481, 67–76 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Williams, R.F.: The structure of Lorenz attractors. Publ. Math. IHES 50, 73–99 (1979)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Willis, A.P.: Openpipe flow: pipe flow code for incompressible flow, technical report. University of Sheffield. www.Openpipeflow.org (2014)
  66. 66.
    Willis, A.P., Cvitanović, P., Avila, M.: Revealing the state space of turbulent pipe flow by symmetry reduction. J. Fluid Mech. 721, 514–540 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Willis, A.P., Short, K.Y., Cvitanović, P.: Symmetry reduction in high dimensions, illustrated in a turbulent pipe. Phys. Rev. E 93, 022204 (2016)ADSCrossRefGoogle Scholar
  68. 68.
    Zammert, S., Eckhardt, B.: Crisis bifurcations in plane Poiseuille flow. Phys. Rev. E 91, 041003 (2015)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Science and Technology (IST)KlosterneuburgAustria
  2. 2.School of Physics, Center for Nonlinear ScienceGeorgia Institute of TechnologyAtlantaUSA

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