Pattern formation: An introduction to methods

Pattern Formation: An Introduction to Methods

Volumn , Issue , 2006, Pages 1-422

  Hoyle, Rebecca B ^a  

^a UNIVERSITY OF SURREY   (United Kingdom)

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[No Author keywords available]

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EID: 84929738454     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1017/CBO9780511616051     Document Type: Book

References (199)

1. Acheson, D. J. (1990) Elementary Fluid Dynamics (Oxford: Clarendon).
2. Agladze, K. I., Panfilov, A.V. And Rudenko, A. N. (1988) Nonstationary rotation of spiral waves: Three-dimensional effect, Physica D. 29, 409–15.
3. Arbell, H. And Fineberg, J. (2002) Pattern formation in two-frequency forced parametric waves, Phys. Rev. E 65, 036224.
4. Arfken, G. B. (2001) Mathematical Methods for Physicists, 5th edition (London: Academic).
5. Arnol’d, V. I. (1983) Geometrical Methods in the Theory of Ordinary Differential Equations (New York: Springer).
6. Aronson, D. G. And Weinberger, H. F. (1978) Multidimensional nonlinear diffusion arising in population genetics, Adv. Math. 30, 33–76.
7. Bar, M., Hildebrand, M., Eiswirth, M., Falcke, M., Engel, H. And Neufeld, M. (1994) Chemical turbulence and standing waves in a surface reaction model: The influence of global coupling and wave instabilities, Chaos 4, 499–508.
8. Barkley, D. (1994) Euclidean symmetry and the dynamics of rotating spiral waves, Phys. Rev. Lett. 72, 164–7.
9. Barkley, D. (1995) Spiral meandering, in Chemical Waves and Patterns, R. Kapral and K. Showalter eds. (Dordrecht: Kluwer).
10. Belmonte, A., Flesselles, J.-M. And Ouyang, Q. (1996) Spiral instability to line sources in forced chemical pattern turbulence, Europhys. Lett. 35, 665–70.
11. Belousov, B. P. (1958) A periodic reaction and its mechanism, Sbornik Referatov po RadiatsionnotMeditsine za 1958 god (Moscow: Medgiz), p.145.
12. Benard, H. (1900) Les tourbillons cellulaires dans une nappe liquide. Premiere partie: Description generale des phenomenes, Rev. Gen. Sci. Pures Ap. 11, 1261–71.
13. Benjamin, T. B. And Feir, J. E. (1967) The disintegration of wave trains on deep water, J. Fluid Mech. 27, 417–30.
14. Bernoff, A. J. (1994) Finite-amplitude convection between stress-free boundaries: Ginzburg-Landau equations and modulation theory, Euro. Jnl. Of Applied Mathematics 5, 267–82.
15. Bestehorn, M. (1993) Phase and amplitude instabilities for Benard-Marangoni convection in fluid layers with large aspect ratio, Phys. Rev. E 48, 3622–34.
16. Bestehorn, M. (1994) Pattern selection in Benard-Marangoni convection, Int. J. Bif. Chaos 4, 1085–94.
17. Billingham, J. And King, A. C. (2000) Wave Motion (Cambridge: Cambridge University Press).
18. Bodenschatz, E., Pesch, W. And Ahlers, G. (2000) Recent developments in Rayleigh-Bénard convection, Annu. Rev. Fluidmech., 32, 709–78.
19. Bodenschatz, E., Pesch, W. And Kramer, L. (1988) Structure and dynamics of dislocations in anisotropic pattern-forming systems, Physica D. 32, 135–45.
20. Bosch Vivancos, I., Chossat, P. And Melbourne, I. (1995) New planforms in systems of partial differential equations with Euclidean symmetry, Arch. Rational Mech. Anal. 131, 199–224.
21. Bressloff, P. C., Cowan, J. D., Golubitsky, M. And Thomas, P. J. (2001A) Scalar and pseudoscalar bifurcations motivated by pattern formation on the visual cortex, Nonlinearity 14, 739–75.
22. Bressloff, P. C., Cowan, J. D., Golubitsky, M., Thomas, P. J. And Wiener, M. C. (2001B) Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex, Phil. Trans. R. Soc. Lond. B 356, 299–330.
23. Brocker, T. And tom Dieck, T. (1985) Representations of Compact Lie Groups (New York: Springer-Verlag).
24. Busse, F. H. (1962) Das Stabilitatsverhalten der Zellularkonvektion bei endlicher Amplitude, Ph.D. Thesis (München: Ludwig-Maximilians-Universitat).
25. Busse, F. H. (1967) The stability of finite amplitude cellular convection and its relation to an extremum principle, J. Fluid Mech. 30, 625–9.
26. Busse, F. H. (1972) The oscillatory instability of convection rolls in a low Prandtl number fluid, J. Fluid Mech. 52, 97–112.
27. Busse, F. H. And Bolton, E. W. (1984) Instabilities of convection rolls with stress-free boundaries near threshold, J. Fluid Mech. 146, 115–25.
28. Busse, F. H. And Clever, R. M. (1979) Instabilities of convection rolls in a fluid of moderate Prandtl number, J. Fluid Mech. 91, 319–35.
29. Busse, F. H. And Heikes, K. E. (1980) Convection in a rotating layer: A simple case of turbulence, Science 208, 173–5.
30. Busse, F. H. And Whitehead, J. A. (1971) Instabilities of convection rolls in a high Prandtl number fluid, J. Fluid Mech., 47, 305–20.
31. Buzano, E. And Golubitsky, M. (1983) Bifurcation involving the hexagonal lattice and the planar Benard problem, Phil. Trans. Roy. Soc. Lond. A 308, 617–67.
32. Cakmur, R. V., Egolf, D. A., Plapp, B. B. And Bodenschatz, E. (1997) Bistability and competition of spatiotemporal chaotic and fixed point attractors in Rayleigh-Benard convection, Phys. Rev. Lett. 79, 1853–6.
33. Carr, J. (1981), Applications of Centre Manifold Theory (New York: Springer-Verlag).
34. Castets, V., Dulos, E., Boissonade, J. And De Kepper, P. (1990) Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern, Phys. Rev. Lett. 64, 2953–6.
35. Chiam, K.-H., Paul, M. R., Cross, M. C. And Greenside, H. S. (2003) Mean flow and spiral defect chaos in Rayleigh-Benard convection, Phys. Rev. E 67, 056206.
36. Chossat, P. And Iooss, G. (1994) The Couette-Taylor Problem (New York: Springer-Verlag).
37. Chossat, P. And Lauterbach, R. (2000) Methods in Equivariant Bifurcations and Dynamical Systems (Singapore: World Scientific).
38. Cicogna, G. (1981) Symmetry breakdown from bifurcations, Lettere al Nuovo Cimento 31, 600–2.
39. Ciliberto, S., Coullet, P., Lega, J., Pampaloni, E. And Peres-Garcia, C. (1990) Defects in roll-hexagon competition, Phys. Rev. Lett. 65, 2370–3.
40. Cohn, P. M. (1982) Algebra, Volume 1, 2nd edn. (Chichester: Wiley).
41. Cornwell, J.F. (1984) Group Theory in Physics, Volume 1 (London: Academic).
42. Coullet, P. And Fauve, S. (1985) Propagative phase dynamics for systems with Galilean invariance. Phys. Rev. Lett. 55, 2857–9.
43. Coullet, P., Gil, L. And Lega, J. (1989) Defect-mediated turbulence, Phys. Rev. Lett. 62, 1619–22.
44. Crawford, J. D. (1994) D4 + T2 mode interactions and hidden rotational symmetry, Nonlinearity 7, 697–739.
45. Crawford, J. D., Golubitsky, M., Gomes, M. G. M., Knobloch, E. And Stewart, I. (1991) Boundary conditions as symmetry constraints, in Singularity Theory and Its Applications, Warwick 1989 (R. M. Roberts and I.N. Stewart) Lecture Notes in Mathematics 2 (Heidelberg: Springer-Verlag).
46. Cross, M. C. (1982) Ingredients of a theory of convective textures close to onset, Phys. Rev. A 25, 1065–76.
47. Cross, M. C. And Hohenberg, P. C. (1993) Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 851–1112.
48. Cross, M. C. And Newell, A. C. (1984) Convection patterns in large aspect ratio systems, Physica D. 10, 299–328.
49. Daniels, K.E. And Bodenschatz, E. (2003) Statistics of defect motion in spatiotemporal chaos in inclined layer convection, Chaos 13, 55–63.
50. Decker, W. And Pesch, W. (1994) Order parameter and amplitude equations for the Rayleigh-Benard convection, J. Phys. Ii France 4, 419–38.
51. Dellnitz, M., Field, M., Golubitsky, M., Hohmann, A. And Jun Ma (1995) ‘Cycling chaos’ IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42, 821–3.
52. Dionne, B. And Golubitsky, M. (1992) Planforms in two and three dimensions, Z. Angew. Math. Phys. 43, 36–62.
53. Dionne, B., Silber, M. And Skeldon, A. C. (1997) Stability results for steady, spatially periodic planforms, Nonlinearity 10, 321–53.
54. Doelman, A., Sandstede, B., Scheel, A. And Schneider, G. (2003) Propagation of hexagonal patterns near onset, Euro. Jnl of App. Math. 14, 85–110.
55. Dubois, M., Berge, P. And Wesfreid, J. (1978) Non Boussinesq convective structures in water near 4° C, J. Phys. (Paris) 39, 1253–7.
56. Echebarria, B. And Perez-Garcia, C. (1998) Phase instabilities in hexagonal patterns, Europhys. Lett. 43, 35–40.
57. Echebarria, B. And Perez-Garcia, C. (2001) Stability of hexagonal patterns in Benard-Marangoni convection, Phys. Rev. E 63, 066307.
58. Echebarria, B. And Riecke, H. (2001) Sideband instabilities and defects of quasipatterns, Physica D. 158, 45–68.
59. Eckhaus, W. (1965) Studies in Nonlinear Stability Theory (Berlin: Springer-Verlag).
60. Edwards, W. S. And Fauve, S. (1993) Parametrically excited quasi-crystalline surface-waves, Phys. Rev. E 47, R788–91.
61. Egolf, D. A., Melnikov, I. V. And Bodenschatz, E. (1998) Importance of local pattern properties in spiral defect chaos, Phys. Rev. Lett. 80, 3228–31.
62. Ercolani, N. M., Indik, R., Newell, A. C. And Passot, T. (2000) The geometry of the phase diffusion equation, J. Nonlinear Sci. 10, 223–74.
63. Ercolani, N. M., Indik, R., Newell, A. C. And Passot, T. (2003) Global description of patterns far from onset: A case study, Physica D. 184, 127–40.
64. Faraday, M. (1831) On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces, Phil. Trans. R. Soc. Lond. 121, 299–340.
65. Fauve, S. (1991) ‘Woods Hole GFD summer school notes: Principal lectures’, Woods Hole Oceanographical Institute Technical Report WHOI-92–16.
66. Fauve, S. (1998) Pattern forming instabilities in Hydrodynamics and Nonlinear Instabilities (eds. C. Godreche and P. Manneville), (Cambridge: Cambridge University Press).
67. Fauve, S., Bolton, E. W. And Brachet, M. E. (1987) Nonlinear oscillatory convection: A quantitative phase dynamics approach, Physica D. 29, 202–14.
68. Fauve, S., Douady, S. And Thual, O. (1991) Drift instabilities of cellular patterns, J. Phys. Ii (Paris) 1, 311–22.
69. FitzHugh, R. (1961) Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445–66.
70. Ginzburg, V. L. And Landau, L. D. (1950) On the theory of superconductivity, Zh. Eksperim. I. Teor. Fiz. 20, 1064–82.
71. Glendinning, P. (1994) Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations (Cambridge: Cambridge University Press).
72. Goldstein, H.F., Knobloch, E. And Silber, M. (1990) Planform selection in rotating convection, Physics of Fluids A 2, 625–7.
73. Golubitsky, M., Marsden J. And Schaeffer, D. (1984) Bifurcation problems with hidden symmetries, in Partial Differential Equations and Dynamical Systems (W. E. Fitzgibbon, ed.), Research Notes in Mathematics 101 (San Francisco: Pitman), 181–210.
74. Golubitsky, M. And Stewart, I. (2002) The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space (Basel: Birkhauser).
75. Golubitsky, M., Stewart, I. And Schaeffer, D. G. (1988) Singularities and Groups in Bifurcation Theory, Volume 2, (New York: Springer-Verlag).
76. Golubitsky, M., Swift, J. W. And Knobloch, E. (1984) Symmetries and pattern selection in Rayleigh-Benard convection, Physica D. 10, 249–76.
77. Gomes, M. G. M., Labouriau, I. S. And Pinho, E. M. (1999) Spatial hidden symmetries in pattern formation, in Pattern Formation in Continuous and Coupled Systems (M. Golubitsky, D. Luss and S. Strogatz, eds.) Springer-Verlag IMA Volumes in Mathematics and its Applications 115, 83–100.
78. Guckenheimer, J. And Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields (New York: Springer-Verlag).
79. Guckenheimer, J. And Holmes, P. (1988) Structurally stable heteroclinic cycles, Math. Proc. Camb. Phil. Soc. 103, 189–92.
80. Hagan, P. S. (1982) Spiral waves in reaction-diffusion equations, SIAM J. Appl. Math. 42, 762–86.
81. Hagan, P. S. And Cohen, M. S. (1981) Diffusion-induced morphogenesis in the development of Dictyostelium, J. Theor. Biol. 93, 881–908.
82. Haken, H. (1978) Synergetics-An Introduction (Berlin: Springer-Verlag).
83. Hendry, M., Nam, K., Guzdar, P. And Ott, E. (2000) Target waves in the complex Ginzburg-Landau equation, Phys. Rev. E 62, 7627–31.
84. Hirsch, M. W. And Smale, S. (1974) Differential Equations, Dynamical Systems and Linear Algebra (New York and London: Academic).
85. Hoyle, R. B. (1993) Long wavelength instabilities of square patterns, Physica D. 67, 198–223.
86. Hoyle, R. B. (1994A) Instabilities of Three-dimensional Patterns, Ph.D. Thesis (Cambridge: University of Cambridge).
87. Hoyle, R. B. (1994B) Phase instabilities of oscillatory standing squares and alternating rolls, Phys. Rev. E 49, 2875–80.
88. Hoyle, R. B. (1995) Nonlinear phase diffusion equations for the longwave instabilities of hexagons, Appl. Math. Lett. 8:3, 81–5.
89. Hoyle, R. B. (1998A) Zigzag and Eckhaus instabilities in a quintic-order nonvariational Ginzburg-Landau equation, Phys. Rev. E 58, 7315–18.
90. Hoyle, R. B. (1998B) Universal instabilities of rolls, squares and hexagons, in Time-dependent nonlinear convection, Adv. Fluid Mech. 19, 51–82. (Southampton: Computer Mechanics).
91. Jahnke, W., Skaggs, W. E. And Winfree, A. T. (1989) Chemical vortex dynamics in the Belousov-Zhabotinsky reaction and in the two-variable Oregonator model, J. Phys. Chem. 93, 740–9.
92. James, G. And Liebeck, M. (1993) Representations and Characters of Groups (Cambridge: Cambridge University Press).
93. Johnston, B. L. And Richman, F. (1997) Numbers and Symmetry (Boca Raton: Crc Press).
94. Kawasaki, K. (1984) Defect-phase dynamics for dissipative media with potential, Prog. Theor. Phys. Suppl. 80, 123–38.
95. Kim, D. T., Kwan, Y., Lee, J. J., Ikeda, T., Uchida, T., Kamjoo, K., Kim, Y.-H., Ong, J. J. C., Athill, C. A., Wu, T.-J., Czer, L., Karagueuzian, H. S. And Chen, P.-S. (1998) Patterns of spiral tip motion in cardiac tissues, Chaos 8, 137–48.
96. Kirk, V. And Silber, M. (1994) A competition between heteroclinic cycles, Nonlinearity 7, 1605–21.
97. Komarova, N. L. And Newell, A. C. (2000) Nonlinear dynamics of sand banks and sand waves, J. Fluid Mech. 415, 285–321.
98. Kramer, L., Ben-Jacob, E., Brand, H. And Cross M. C. (1982) Wavelength selection in systems far from equilibrium, Phys. Rev. Lett. 49, 1891–4.
99. Kramer, L. And Zimmermann W. (1985) On the Eckhaus instability for spatially periodic patterns, Physica D 16, 221–32.
100. Krishnamurti, R. (1970) On the transition to turbulent convection. Part 2: The transition to time-dependent flow, J. Fluid Mech. 42, 309–20.
101. Krishnamurti, R. (1973) Some further studies on the transition to turbulent convection, J. Fluid Mech. 60, 285–303.
102. Kudrolli, A., Pier B. And Gollub, J. P. (1998) Superlattice patterns in surface waves, Physica D 123, 99–111.
103. Kuecken, M. U. (2004) On the Formation of Fingerprints, Ph.D. Thesis (Tucson: University of Arizona).
104. Kumar, K. (1996) Linear theory of Faraday instability in viscous liquids, Proc. R. Soc. Lond.A 452, 1113–26.
105. Kuppers, G. And Lortz, D. (1969) Transition from laminar convection to thermal turbulence in a rotating fluid layer, J. Fluid Mech. 35, 609–20.
106. Kuramoto, Y. (1984A) Phase dynamics of weakly unstable periodic structures, Prog. Theor. Phys. 71, 1182–96.
107. Kuramoto, Y. (1984B) Chemical Oscillations, Waves and Turbulence (New York: Springer-Verlag).
108. Kuramoto, Y. And Tsuzuki, T. (1976) Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys. 55, 356–69.
109. Kuske, R. And Milewski, P. (1999) Modulated two-dimensional patterns in reaction-diffusion systems, Euro. Jnl of App. Math. 10, 157–84.
110. Kuznetsov, E. A., Nepomnyashchy, A. A. And Pismen, L. M. (1995) New amplitude equation for Boussinesq convection and nonequilateral hexagonal patterns, Phys. Lett. A 205, 261–5.
111. Lauzeral, J., Metens, S. And Walgraef, D. (1993) On the phase dynamics of hexagonal patterns, Europhys. Lett. 24, 707–12.
112. Lee, K. J. (1997) Wave pattern selection in an excitable system, Phys. Rev. Lett. 79, 2907–10.
113. Lee, K. J., Cox, E. C. And Goldstein, R. E. (1996) Competing patterns of signaling activity in Dictyostelium discoideum, Phys. Rev. Lett. 76, 1174–7.
114. Li, G., Ouyang, Q., Petrov, V. And Swinney, H.L. (1996) Transition from simple rotating chemical spirals to meandering and traveling spirals, Phys. Rev. Lett. 77, 2105–8.
115. Malomed, B. A., Nepomnyaschchi I. A. A. And Tribelskii, M. I. (1989) Two-dimensional quasiperiodic structures in nonequilibrium systems, Sov. Phys. Jetp 69, 388–96.
116. Malomed, B. A., Nepomnyashchy, A. A. And Tribelsky, M. I. (1990) Domain boundaries in convection patterns, Phys. Rev. A 42, 7244–63.
117. Manneville, P. (1990) Dissipative structures and weak turbulence (San Diego: Academic).
118. Manoel, M. And Stewart, I. (2000) The classification of bifurcations with hidden symmetries, P Lond. Math. Soc. 80, 198–234.
119. Matthews, P. C. (2003) Pattern formation on a sphere, Phys. Rev. E 67, 036206.
120. Matthews, P. C. (2004) Automating symmetry-breaking calculations, LMS J. Comp. & Math. 7, 101–19.
121. Matthews, P. C. And Cox, S. M. (2000A) One-dimensional pattern formation with Galilean invariance near a stationary bifurcation. Phys. Rev. E 62, R1473–6.
122. Matthews, P. C. And Cox, S. M. (2000B) Pattern formation with a conservation law. Nonlinearity 13, 1293–320.
123. May, R. M. And Leonard, W. J. (1975) Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29, 243–53.
124. Melbourne, I. (1998) Derivation of the time-dependent Ginzburg-Landau equation on the line, J. Nonlinear Sci. 8, 1–15.
125. Melbourne, I. (1999) Steady-state bifurcation with Euclidean symmetry, T. Am. Math. Soc. 351, 1575–603.
126. Melbourne, I., Chossat, P. And Golubitsky, M. (1989) Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry, Proc. Roy. Soc. Edin. 113A, 315–45.
127. Melnikov, I. V., Egolf, D. A., Jeanjean, S., Plapp, B. B. And Bodenschatz, E. (2000) Invasion of spiral defect chaos into straight rolls in Rayligh-Benard convection, in Stochastic Dynamics and Pattern Formation in Biological and Complex Systems (eds. S. Kim, K. J. Lee, T. K. Lim and W. Sung), AIP Conference Proceedings501(1), pp. 36–42.
128. Melo, F., Umbanhowar, P. B. And Swinney, H. L. (1995) Hexagons, kinks and disorder in oscillated granular layers, Phys. Rev. Lett. 75, 3838–41.
129. Mielke, A. (1997) Mathematical analysis of sideband instabilities with application to Rayleigh-Benard convection, J. Nonlinear Sci. 7, 57–99.
130. Morris, S. W., Bodenschatz, Cannell, D. S. And Ahlers, G. (1993) Spiral defect chaos in large aspect ration Rayleigh-Benard convection, Phys. Rev. Lett. 71, 2026–9.
131. Mullin, T. (2002) Mixing and de-mixing, Science 295, 1851.
132. Nagumo, J., Arimoto, S. And Yoshizawa, S. (1962) An active pulse transmission line simulating nerve axon, Proc. Ire 50, 2061–70.
133. Nagy-Ungvarai, Z., Ungvarai, J. And Muller, S. C. (1993) Complexity in spiral wave dynamics, Chaos 3, 15–19.
134. Nettesheim, S., von Oertzen, A., Rotermund, H. H. And Ertl, G. (1993) Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110): Front propagation and spiral waves, J. Chem. Phys. 98, 9977–85.
135. Newell, A. C., Passot, T., Bowman, C., Ercolani, N. And Indik, R. (1996) Defects are weak and self-dual solutions of the Cross-Newell phase diffusion equation for natural patterns, PhysicaD. 97, 185–205.
136. Newell, A. C., Passot, T. And Souli, M. (1990) The phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers, J. Fluid Mech. 220, 187–252.
137. Newell, A. C. And Whitehead, J. A. (1969) Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38, 279–303.
138. Norbury, J., Wei, J. And Winter, M. (2002) Existence and stability of singular patterns in a Ginzburg-Landau equation coupled with a mean field, Nonlinearity 15, 2077–96.
139. Ondarcuhu, T., Mindlin, G. B., Mancini, H. L. And Perez García, C. (1993) Dynamical patterns in Benard-Marangoni convection in a square container, Phys. Rev. Lett. 70, 3892–5.
140. Ouyang, Q. And Flesselles, J.-M. (1996) Transition from spirals to defect turbulence driven by a convective instability, Nature 379, 143–6.
141. Ouyang, Q., Swinney, H. L. And Li, G. (2000) Transition from spirals to defect-mediated turbulence driven by a Doppler instability, Phys. Rev. Lett. 84, 1047–50.
142. Panfilov, A. V. (1998) Spiral breakup as a model of ventricular fibrillation, Chaos 8, 57–64.
143. Passot, T. And Newell, A. C. (1994) Towards a universal theory for natural patterns, Physica D. 74, 301–52.
144. Pearson, J. R. A. (1958) On convection cells induced by surface tension, J. Fluid Mech. 4, 489–500.
145. Perez-Munuzuri, V., Aliev, R., Vasiev, B., Perez-Villar, V. And Krinsky, V. I. (1991) Super-spiral structures in an excitable medium, Nature 353, 740–2.
146. Pismen, L. M. (1981) Bifurcation into wave patterns and turbulence in reaction-diffusion equations, Phys. Rev. A 23, 334–44.
147. Plapp, B. B., Egolf, D. A., Bodenschatz, E. And Pesch, W. (1998) Dynamics and selection of giant spirals in Rayleigh-Benard convection, Phys. Rev. Lett. 81, 5334–7.
148. Plesser, T., Muller, S. C. And Hess, B. (1990) Spiral wave dynamics as a function of proton concentration in the ferroin-catalyzed Belousov-Zhabotinskii reaction, J. Phys. Chem. 94, 7501–7.
149. Pocheau, A. And Croquette, V. (1984) Dislocation motion: A wavenumber selection mechanism in Rayleigh-Benard convection, J. Physique 45, 35–48.
150. Pollicott, S. L., Matthews, P. C. And Cox, S. M. (2003) Instability of convection in a fluid layer rotating about an oblique axis, Phys. Rev. E 67, 016301.
151. Pomeau, Y. And Manneville, P. (1979) Stability and fluctuations of a spatially periodic flow, J. Physique Lett. 40, L609–12.
152. Pomeau, Y. And Zaleski, S. (1983) Pattern selection in a slowly varying environment. J. Physique Lett. 44, L135–41.
153. Pomeau, Y., Zaleski, S. And Manneville, P. (1983) Dislocation motion in cellular structures. Phys. Rev. A 27, 2710–26.
154. Ponty, Y., Passot, T. And Sulem, P. L. (1997) Pattern dynamics in rotating convection at finite Prandtl number, Phys. Rev. E. 56, 4162–78.
155. Porter, D. And Stirling, D. S. G. (1990) Integral Equations: A Practical Treatment from Spectral Theory to Applications (Cambridge: Cambridge University Press).
156. Proctor, M. R. E. (2001) Finite amplitude behaviour of the Matthews-Cox instability, Phys. Lett. A 292, 181–7.
157. Proctor, M. R. E. And Jones, C. A. (1988) The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1:2 resonance, J. Fluid Mech. 188, 301–35.
158. Rayleigh, Lord (1916) On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Philos. Mag. 32, 529–46.
159. Rucklidge, A. M. (2003) Pattern formation in large domains, Phil. Trans. R. Soc. Lond. A 361, 2649–64.
160. Van Saarloos, W. (2003) Front propagation into unstable states, Phys. Rep. 386, 29–222.
161. Sandstede, B. And Scheel, A. (2000) Absolute versus convective instability of spiral waves, Phys. Rev. E 62, 7708–14.
162. Sandstede, B. And Scheel, A. (2001) Superspiral structures of meandering and drifting spiral waves, Phys. Rev. Lett. 86, 171–4.
163. Sandstede, B. And Scheel, A. (2004) Defects in oscillatory media: Towards a classification. Siam J. Applied Dynamical Systems 3, 1–68.
164. Sandstede, B., Scheel, A. And Wulff, C. (1997) Center-manifold reduction for spiral waves, C.R. Acad. Sci., Serie I, Math. 324, 153–8.
165. Sandstede, B., Scheel, A. And Wulff, C. (1999A) Dynamical behavior of patterns with Euclidean symmetry, in Pattern Formation in Continuous and Coupled Systems (M. Golubitsky, D. Luss and S. Strogatz, eds.) Springer-Verlag, IMA Volumes in Mathematics and its Applications115, 249–64.
166. Sandstede, B., Scheel, A. And Wulff, C. (1999B) Bifurcations and dynamics of spiral waves. J. Nonlinear Sci. 9, 439–78.
167. Schlüter, A., Lortz, D. And Busse, F. H. (1965) On the stability of steady finite amplitude convection, J. Fluid Mech. 23, 129–44.
168. Schrödinger, E. (1926A) Quantisierung als Eigenertproblem (Part 1), Annalen der Physik 79, 361–76.
169. Schrödinger, E. (1926B) Über das Verhaltnis der Heisenberg-Born-Jordanschen Quantenmechanik zur der meinen, Annalen der Physik 79, 734–55.
170. Schrödinger, E. (1926C) Quantisierung als Eigenertproblem (Part 3) Annalen der Physik 80, 437–90.
171. Schrödinger, E. (1926D) Quantisierung als Eigenertproblem (Part 4) Annalen der Physik 81, 109–39.
172. Segel, L. A. (1969) Distant side-walls cause slow amplitude modulation of cellular convection, J. Fluid Mech. 38, 203–24.
173. Shechtman, D., Blech, I., Gratias, D. And Cahn, J. W. (1984) Metallic phase with long-range orientational order and no translational symmetry, Phy. Rev. Lett. 53, 1951–3.
174. Siggia, E. D. And Zippelius, A. (1981) Dynamics of defects in Rayleigh-Benard convection, Phys. Rev. A 24, 1036–49.
175. Silber, M. And Proctor, M. R. E. (1998) Nonlinear competition between small and large hexagonal patterns, Phys. Rev. Lett. 81, 2450–3.
176. Sivashinsky, G. I. (1977) Nonlinear analysis of hydrodynamic instability in laminar flames. Part I. Derivation of basic equations, Acta Astronautica 4, 1177–206.
177. Skinner, G. S. And Swinney, H.L. (1991) Periodic to quasiperiodic transition of chemical spiral rotation, Physica D. 48, 1–16.
178. Steeb, W.-H. (1996) Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra (Singapore: World Scientific).
179. Sushchik, M. M. And Tsimring, L. S. (1994) The Eckhaus instability in hexagonal patterns, Physica D 74, 90–106.
180. Swift, J. And Hohenberg, P. C. (1977) Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15, 319–28.
181. Tesauro, G. And Cross, M. C. (1986) Climbing of dislocations in nonequilibrium patterns, Phys. Rev. A 34, 1363–79.
182. Tesauro, G. And Cross, M. C. (1987) Grain boundaries in models of convective patterns, Philos. Mag. A 56, 703–24.
183. Tse, D. P, Rucklidge, A. M., Hoyle, R. B. And Silber, M. (2000) Spatial period-multiplying instabilities of hexagonal Faraday waves, Physica D. 146, 367–87.
184. Tuckerman, L. S. And Barkley, D. (1990) Bifurcation analysis of the Eckhaus instability, Physica D. 46, 57–86.
185. Turing, A. M. (1952) The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B 237, 37–72.
186. Tveitereid, M. And Palm, E. (1976) Convection due to internal heat sources, J. Fluid Mech. 76, 481–99.
187. Vanderbauwhede, A. (1980) Local bifurcation and symmetry, Habilitation thesis (Gent: Rijksuniversiteit).
188. Vasiev, B., Siegert, F. And Weijer, C. (1997) Multiarmed spirals in excitable media, Phys. Rev. Lett. 78, 2489–92.
189. Walter, Th., Pesch, W. And Bodenschatz, E. (2004) Dislocation dynamics in Rayleigh-Benard convection, Chaos 14, 933–9.
190. White, D. B. (1988) The planforms and onset of convection with a temperature-dependent viscosity, J. Fluid Mech. 191, 247–86.
191. Willis, G. E. And Deardorff, J. W. (1970) The oscillatory motions of Rayleigh convection, J. Fluid Mech. 44, 661–72.
192. Winfree, A. T. (1973) Scroll-shaped waves of chemical activity in three dimensions, Science 181, 937–9.
193. Winfree, A. T. (1998) Evolving perspectives during 12 years of electrical turbulence, Chaos 8, 1–19.
194. Wulff, C. (2000) Transitions from relative equilibria to relative periodic orbits. Documenta Math. 5, 227–74.
195. Wulff, C. (2002) Spiral waves and Euclidean symmetries. Z. Phys. Chem. 216, 535–50.
196. Yang, L., Dolnik, M., Zhabotinsky, A. M. And Epstein, I. R. (2002) Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes, Phys. Rev. Lett. 88, 208–303.
197. Zaikin, A. And Zhabotinsky, A. M. (1970) Concentration wave propagation in a two-dimensional liquid-phase self-oscillating system, Nature 225, 535–7.
198. Zhou, L. Q. And Ouyang, Q. (2000) Experimental studies on long-wavelength instability and spiral breakup in a reaction-diffusion system, Phys. Rev. Lett. 85, 1650–3.
199. Zippelius, A. And Siggia, E. D. (1983) Stability of finite-amplitude convection, Phys. Fluids 26, 2905–15.