Pattern Formation for the Swift-Hohenberg Equation on the Hyperbolic Plane

  • Pascal Chossat
  • Grégory Faye


In this paper we present an overview of pattern formation analysis for an analogue of the Swift-Hohenberg equation posed on the real hyperbolic space of dimension two, which we identify with the Poincaré disc \(\mathbb{D }\). Different types of patterns are considered: spatially periodic stationary solutions, radial solutions and traveling waves, however there are significant differences in the results with the Euclidean case. We apply equivariant bifurcation theory to the study of spatially periodic solutions on a given lattice of \(\mathbb{D }\) also called H-planforms in reference with the “planforms” introduced for pattern formation in Euclidean space. We consider in details the case of the regular octagonal lattice and give a complete descriptions of all H-planforms bifurcating in this case. For radial solutions (in geodesic polar coordinates), we present a result of existence for stationary localized radial solutions, which we have adapted from techniques on the Euclidean plane. Finally, we show that unlike the Euclidean case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf bifurcation to traveling waves which are invariant along horocycles of \(\mathbb{D }\) and periodic in the “transverse” direction. We highlight our theoretical results with a selection of numerical simulations.


Swift-Hohenberg equation Pattern formation Poincaré disk Equivariant bifurcation Traveling wave 



GF is grateful to James Rankin and David Lloyd for their helpful comments on the functionality of AUTO.


  1. 1.
    Allaire, G.: Analyse numérique et optimisation. Éd. de l’Ecole Polytechnique (2005)Google Scholar
  2. 2.
    Anker, J.P., Pierfelice, V.: Nonlinear schrödinger equation on real hyperbolic spaces. In Annales de l’Institut Henri Poincare (C) Non Linear Analysis, volume 26, pp. 1853–1869. Elsevier (2009)Google Scholar
  3. 3.
    Anker, J.P., Pierfelice, V., Vallarino, M.: The wave equation on hyperbolic spaces. J. Differ. Equ. 252, 4392–4409 (2012)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Aurich, R., Steiner, F.: Periodic-orbit sum rules for the hadamard-gutzwiller model. Phys. D 39, 169–193 (1989)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Aurich, R., Steiner, F.: Statistical properties of highly excited quantum eigenstates of a strongly chaotic system. Phys. D 64, 185–214 (1993)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Avitabile, D., Lloyd, D.J.B., Burke, J., Knobloch, E., Sandstede, B.: To snake or not to snake in the planar swift-hohenberg equation. SIAM J. Appl. Dyn. Syst. 9, 704 (2010)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Balazs, N.L., Voros, A.: Chaos on the pseudosphere. Phys. Rep. 143(3), 109–240 (1986)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Banica, V.: The nonlinear schrödinger equation on hyperbolic space. Commun. Partial Differ. Equ. 32(10), 1643–1677 (2007)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Broughton, S.A.: Classifying finite group actions on surfaces of low genus. J. Pure Appl. Algebra 69(3), 233–270 (1991)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Broughton, S.A., Dirks, R.M., Sloughter, M.T., Vinroot, C.R.: Triangular surface tiling groups for low genus. Technical report, MSTR (2001)Google Scholar
  11. 11.
    Burke, J., Knobloch, E.: Localized states in the generalized swift-hohenberg equation. Phys. Rev. E 73(5), 056211 (2006)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Burke, J., Knobloch, E.: Homoclinic snaking: structure and stability. Chaos 17(3), 7102 (2007)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Burke, J., Knobloch, E.: Snakes and ladders: localized states in the swift-hohenberg equation. Phys. Lett. A 360(6), 681–688 (2007)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Buser, P.: Geometry and Spectra of Compact Riemann Surfaces, vol. 106. Springer, Berlin (1992)Google Scholar
  15. 15.
    Chossat, P., Lauterbach, R.: Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific Publishing Company, River Edge (2000)CrossRefMATHGoogle Scholar
  16. 16.
    Chossat, P., Lauterbach, R., Melbourne, I.: Steady-state bifurcation with 0 (3)-symmetry. Arch. Ration. Mech. Anal. 113(4), 313–376 (1990)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Chossat, P., Faugeras, O.: Hyperbolic planforms in relation to visual edges and textures perception. Plos. Comput. Biol. 5(12), e1000625 (2009)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Chossat, P., Faye, G., Faugeras, O.: Bifurcations of hyperbolic planforms. J. Nonlinear Sci. 21(4), 465–498 (2011)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Ciarlet, P.G., Lions, J.L. (eds.): Handbook of Numerical Analysis. Vol. ii. Finite Element Methods (Part1). North-Holland, Amsterdam (1991)Google Scholar
  20. 20.
    Collet, P., Eckmann, J.-P.: Space-time behaviour in problems of hydrodynamic type: a case study. Nonlinearity 5, 1265–1302 (1992)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Collet, P.: Thermodynamic limit of the Ginzburg-Landau equations. Nonlinearity 7, 1175–1190 (1994)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Cornish, N.J., Spergel, D.N.: On the eigenmodes of compact hyperbolic 3-manifolds. Technical report, arXiv (1999)Google Scholar
  23. 23.
    Cornish, N.J., Turok, N.G.: Ringing the eigenmodes from compact manifolds. Technical report, arXiv (1998)Google Scholar
  24. 24.
    Dias, F., Iooss, G.: Water-waves as a spatial dynamical system. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, Chap. 10, pp. 443–499. Elsevier, Amsterdam (2003)CrossRefGoogle Scholar
  25. 25.
    Dionne, B., Golubitsky, M.: Planforms in two and three dimensions. ZAMP 43, 36–62 (1992)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Dionne, B., Silber, M., Skeldon, A.C.: Stability results for steady, spatially periodic planforms. Nonlinearity 10, 321 (1997)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Doedel, E.J., Oldeman, B.: Continuation and bifurcation software for ordinary differential equations. Technical report (2009)Google Scholar
  28. 28.
    Erdelyi, : Higher Transcendental Functions, vol. 1. Robert E. Krieger Publishing Company, Malabar (1985)Google Scholar
  29. 29.
    Faye, G.: Reduction method for studying localized solutions of neural field equations on the poincaré disk. Comptes Rendus de l’Académie des Sciences, Mathématique (2012)Google Scholar
  30. 30.
    Faye, G., Chossat, P.: A spatialized model of textures perception using structure tensor formalism. AIMS J. Netw. Heterog. Media 10(1), 211–260 (2013)MathSciNetGoogle Scholar
  31. 31.
    Faye, G., Chossat, P.: Bifurcation diagrams and heteroclinic networks of octagonal H-planforms. J. Nonlinear Sci. 22(3), 49 (2011)MathSciNetGoogle Scholar
  32. 32.
    Faye, G., Chossat, P., Faugeras, O.: Analysis of a hyperbolic geometric model for visual texture perception. J. Math. Neurosci. 1(1), 4 (2011)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Faye, G., Rankin, J., Lloyd, D.J.: Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disk. Nonlinearity 26, 437–478 (2013)CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    GAP: Groups, Algorithms and Programming. URL of GAPGoogle Scholar
  35. 35.
    Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, Vol. II. Springer, Berlin (1988)CrossRefGoogle Scholar
  36. 36.
    Haragus, M., Iooss, G.: Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Systems. EDP Sci. Springer UTX series, Berlin (2010)Google Scholar
  37. 37.
    Hartshorne, R.: Algebraic Geometry, vol. 52. Springer, Berlin (1977)MATHGoogle Scholar
  38. 38.
    Helgason, S.: Groups and Geometric Analysis, Vol. 83 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2000)CrossRefGoogle Scholar
  39. 39.
    Hoyle, R.B.: Pattern Formation: An Introduction to Methods. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  40. 40.
    Inoue, K.T.: Computation of eigenmodes on a compact hyperbolic 3-space. Technical report, arXiv (1999)Google Scholar
  41. 41.
    Iooss, G., Adelmeyer, M.: Topics in Bifurcation Theory and Applications, Vol. 3 of Advanced Series in Nonlinear Dynamics. World Scientific, Singapore (1998)Google Scholar
  42. 42.
    Iooss, G., Peroueme, M.C.: Perturbed homoclinic solutions in reversible 1:1 resonance vector fields. J. Differ. Equ. 102(1), 62–88 (1993)CrossRefMathSciNetMATHGoogle Scholar
  43. 43.
    Iooss, G., Rucklidge, A.M.: On the existence of quasipattern solutions of the swift-hohenberg equation. J. Nonlinear Sci. 20(3), 361–394 (2010)CrossRefMathSciNetMATHGoogle Scholar
  44. 44.
    Iwaniec, H.: Spectral Methods of Automorphic Forms, vol. 53 of AMS Graduate Series in Mathematics. AMS Bookstore, Boston (2002)Google Scholar
  45. 45.
    Katok, S.: Fuchsian Groups. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago (1992)Google Scholar
  46. 46.
    Kirchgassner, K.: Nonlinearly resonant surface waves and homoclinic bifurcation. Adv. Appl. Mech. 26, 135–181 (1988)CrossRefMathSciNetGoogle Scholar
  47. 47.
    Kirchgassner, K., Kielhofer, H.: Stability and bifurcation in fluid dynamics. Rocky Mountain J. Math 3, 275–318 (1973)CrossRefMathSciNetGoogle Scholar
  48. 48.
    Landau, L.D., Ginzburg, V.L.: On the theory of superconductivity. J. Exp. Theor. Phys. (USSR) 20, 1064 (1950)Google Scholar
  49. 49.
    Lang, S.: Algebra, 3rd edn. Addison-Wesley, Boston (1993)MATHGoogle Scholar
  50. 50.
    Lehoucq, R., Weeks, J., Uzan, J-P., Gausmann, E., Luminet, J-P.: Eigenmodes of 3-dimensional spherical spaces and their application to cosmology. Technical report, arXiv (2002)Google Scholar
  51. 51.
    Lloyd, D., Sandstede, B.: Localized radial solutions of the swift-hohenberg equation. Nonlinearity 22, 485 (2009)CrossRefMathSciNetMATHGoogle Scholar
  52. 52.
    Lloyd, D.J.B., Sandstede, B., Avitabile, D., Champneys, A.R.: Localized hexagon patterns of the planar swift-hohenberg equation. SIAM J. Appl. Dyn. Syst. 7(3), 1049–1100 (2008)CrossRefMathSciNetMATHGoogle Scholar
  53. 53.
    McCalla, S.: Localized Structures in the Multi-dimensional Swift-Hohenberg Equation. PhD thesis, Brown UNiversity (2011)Google Scholar
  54. 54.
    McCalla, S., Sandstede, B.: Snaking of radial solutions of the multi-dimensional swift-hohenberg equation: a numerical study. Phys. D: Nonlinear Phenom. 239(16), 1581–1592 (2010)CrossRefMathSciNetMATHGoogle Scholar
  55. 55.
    McCalla, S., Sandstede, B.: Spots in the Swift-Hohenberg Equation, Preprint (2012)Google Scholar
  56. 56.
    Melbourne, I.: A singularity theory analysis of bifurcation problems with octahedral symmetry. Dyn. Stab. Syst. 1(4), 293–321 (1986)CrossRefMathSciNetMATHGoogle Scholar
  57. 57.
    Miller, W.: Symmetry Groups and Their Applications. Academic Press, New York (1972)MATHGoogle Scholar
  58. 58.
    Pollicott, M.: Distributions at infinity for riemann surfaces. In: Stefan Banach Center, editor, Dynamical Systems and Ergodic Theory 23, 91–100 (1989)Google Scholar
  59. 59.
    Scheel, A.: Bifurcation to spiral waves in reaction-diffusion systems. SIAM J. Math. Anal. 29(6), 1399–1418 (1998)CrossRefMathSciNetMATHGoogle Scholar
  60. 60.
    Scheel, A.: Radially Symmetric Patterns of Reaction-Diffusion Systems, vol. 165. American Mathematical Society, Providence (2003)Google Scholar
  61. 61.
    Schmit, C.: Quantum and classical properties of some billiards on the hyperbolic plane. In: Chaos and Quantum Physics, M.-J. Giannoni (eds.), Elsevier, New York, pp. 335-369 (1991)Google Scholar
  62. 62.
    Series, C.: Some geometrical models of chaotic dynamics. Proc R Soc Lond. Ser. A Math. Phys. Sci. 413(1844), 171–182 (1987)CrossRefMathSciNetMATHGoogle Scholar
  63. 63.
    Swift, J., Hohenberg, P.C.: Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15(1), 319 (1977)CrossRefGoogle Scholar
  64. 64.
    Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. R. Soc. Lond. Ser. A 223, 289–343 (1923)CrossRefMATHGoogle Scholar
  65. 65.
    Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B Biol. Sci. 237(641), 37–72 (1952)CrossRefGoogle Scholar
  66. 66.
    Virchenko, N.O., Fedotova, I.: Generalized Associated Legendre Functions and Their Applications. World Scientific Pub Co Inc, Singapore (2001)CrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.JAD LaboratoryUniversity of Nice Sophia Antipolis & CNRSNice Cedex 02France
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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