Spatio-Temporal Dynamics of Reaction-Diffusion Patterns

  • Bernold Fiedler
  • Arnd Scheel


In this survey we look at parabolic partial differential equations from a dynamical systems point of view. With origins deeply rooted in celestial mechanics, and many modern aspects traceable to the monumental influence of Poincaré, dynamical systems theory is mainly concerned with the global time evolution T(t)u 0 of points u 0 — and of sets of such points — in a more or less abstract phase space X. The success of dynamical concepts such as gradient flows, invariant manifolds, ergodicity, shift dynamics, etc. during the past century has been enormous — both as measured by achievement, and by vitality in terms of newly emerging questions and long-standing open problems.


Hopf Bifurcation Global Attractor Essential Spectrum Spiral Wave Morse Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AbRo67]
    R. Abraham and J. Robbin. Transversal Mappings and Flows. Benjamin Inc., Amsterdam, 1967.zbMATHGoogle Scholar
  2. [A179]
    N. Alikakos. An application of the invariance principle to reaction diffusion equations. J. Diff. Eqns. 33 (1979), 201–225.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [Ala189]
    N. Alikakos, P.W. Bates, and G. Fusco. Slow motion for the Cahn-Hilliard equation in one space dimension. Preprint (1989).Google Scholar
  4. [A1Ge90]
    E.L. Allgower and K. Georg. Numerical Continuation Methods. An Introduction. Springer-Verlag, Berlin, 1990.zbMATHCrossRefGoogle Scholar
  5. [An86]
    S. Angenent. The Morse-Smale property for a semi-linear parabolic equation. J. Diff. Eqns. 62 (1986), 427–442.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [An88]
    S. Angenent. The zero set of a solution of a parabolic equation. Grelle J. reine angew. Math., 390 (1988), 79–96.MathSciNetzbMATHGoogle Scholar
  7. [An90]
    S. Angenent. Parabolic equations for curves on surfaces. I: curves with p-integrable curvature. Ann. Math. 132 (1990), 451–483.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [An91]
    S. Angenent. Parabolic equations for curves on surfaces. II: Intersections, blow-up and generalized solutions. Ann. Math., 133 (1991), 171–215.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [An93]
    S. Angenent. A variational interpretation of Melnikov’s function and exponentially small separatrix splitting. Lond. Math. Soc. Lect. Note Ser., 192 (1993), 5–35.Google Scholar
  10. [AnFi88]
    S. Angenent and B. Fiedler. The dynamics of rotating waves in scalar reaction diffusion equations. Trans. Amer. Math. Soc., 307 (1988), 545–568.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [Ana187]
    S. Angenent, J. Mallet-Paret, and L.A. Peletier. Stable transition layers in a semilinear boundary value problem. J. Diff. Eqns. 67 (1987), 212–242.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [An91]
    D.V. Anosov. Dynamical Systems with Hyperbolic Behaviour. Enc. Math. Sc. 66, Dynamical Systems IX. Springer-Verlag, New York, 1991.Google Scholar
  13. [AnAr88]
    D. V. Anosov and V. I. Arnol’d. Ordinary differential equations and smooth dynamical systems. Enc. Math. Sc. 1, Dynamical Systems I. Springer-Verlag, Berlin, 1988.CrossRefGoogle Scholar
  14. [Ara192]
    I. S. Aranson, L. Aranson, L. Kramer, and A. Weber. Stability limits of spirals and travelling waves in nonequilibrium media. Phys. Rev. A 46 (1992), 2992–2995.CrossRefGoogle Scholar
  15. [Ara194]
    I.S. Aranson, L. Kramer, and A. Weber. Core instability and spatiotemporal intermittency of spiral waves in oscillatory media. Phys. Rev. Lett. 72, 2316 (1994).CrossRefGoogle Scholar
  16. [Ar92]
    V.I. Arnol’d. Theory of Bifurcations and Catastrophes. Enc. Math. Sc. 5, Dynamical Systems V. Springer-Verlag, Berlin, 1992.Google Scholar
  17. [Ar93]
    V.I. Arnol’d. Singularity Theory I. Enc. Math. Sc. 6, Dynamical Systems VI. Springer-Verlag, New York, 1993.Google Scholar
  18. [Ar94a]
    V.I. Arnol’d. Bifurcation Theory and Catastrophe Theory. Enc. Math. Sc. 5, Dynamical Systems V. Springer-Verlag, New York, 1994.Google Scholar
  19. [Ar94b]
    V.I. Arnol’d. Singularity theory II, Applications. Enc. Math. Sc. 8, Dynamical Systems VIII. Springer-Verlag, New York, 1993.Google Scholar
  20. [Ara185]
    V.I. Arnol’d, S.M. Gusejn-Zade, and A.N. Varchenko. Singularities of Differentiable Maps. Volume I: The Classification of Critical points, Caustics and Wave Fronts. Birkhäuser, Boston, 1985.Google Scholar
  21. [Ara188]
    V.I. Arnol’d, V.V. Kozlov, and A.I. Neishtadt. Mathematical Aspects of Classical and Celestial Mechanics. Enc. Math. Sc. 3, Dynamical Systems III. Springer-Verlag, New York, 1988.CrossRefGoogle Scholar
  22. [ArNo90]
    V.I. Arnol’d and S.P. Novikov. Symplectic Geometry and its Applications. Enc. Math. Sc. 4, Dynamical Systems IV. Springer-Verlag, New York, 1990.Google Scholar
  23. [ArNo94]
    V.I. Arnol’d and S.P. Novikov. Integrable Systems. Nonholonomic Dynamical Systems. Enc. Math. Sc. 16, Dynamical Systems VII. Springer-Verlag, New York, 1994.Google Scholar
  24. [ArVi89]
    V.I. Arnol’d and M.I. Vishik et al. Some solved and unsolved problems in the theory of differential equations and mathematical physics. Russian Math. Surveys, 44 (1989), 157–171.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [AM97]
    P. Ashwin and I. Melbourne. Noncompact drift for relative equilibria and relative periodic orbits. Nonlinearity, 10 (1997), 595–616.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [Asa199]
    P. Ashwin, I. Melbourne, and M. Nicol. Drift bifurcations of relative equilibria and transitions of spiral waves. Nonlinearity 12 (1999), 741–755.Google Scholar
  27. [Asa101]
    P. Ashwin, I. Melbourne, and M. Nicol. Hypermeander of spirals: local bifurcations and statistical properties. Phys. D 156 (2001), 364–382.Google Scholar
  28. [BaVi92]
    A.V. Babin and M.I. Vishik. Attractors of Evolution Equations. North Holland, Amsterdam, 1992.zbMATHGoogle Scholar
  29. [BäEi93]
    M. Bär and M. Eiswirth. Turbulence due to spiral breakup in a continuous excitable medium. Phys. Rev. E 48 (1993), 1635–1637.Google Scholar
  30. [BäOr99]
    M. Bär and M. Or-Guil. Alternative scenarios of spiral breakup in a reaction-diffusion model with excitable and oscillatory dynamics. Phys. Rev. Lett. 82 (1999), 1160–1163.CrossRefGoogle Scholar
  31. [Ba92]
    D. Barkley. Linear stability analysis of rotating spiral waves in excitable media. Phys. Rev. Lett. 68 (1992), 2090–2093.CrossRefGoogle Scholar
  32. [Ba93]
    D. Barkley. Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett. 72 (1994), 164–167.CrossRefGoogle Scholar
  33. [Bar95]
    D. Barkley. Spiral meandering. In R. Kapral and K. Showalter (eds.), Chemical Waves and Patterns, p.163–190, Kluwer, 1995.Google Scholar
  34. [Baa193]
    G. Barles, H.M. Soner, and P.E. Souganidis. Front propagation and phase field theory. SIAM J. Contr. Optim. 31 (1993), 439–469.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [Bea197]
    A. Belmonte, J.-M. Flesselles, and Q. Ouyang. Experimental Survey of Spiral Dynamics in the Belousov-Zhabotinsky Reaction. J. Physique II 7 (1997), 1425–1468.Google Scholar
  36. [BeNi90]
    H. Berestycki and L. Nirenberg. Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains. Coll. Analysis, et cetera, 115–164, Academic Press Boston, 1990.Google Scholar
  37. [BePh96]
    I. Berkes and W. Philipp. Trigonometric series and uniform distribution mod 1. Stud. Sci. Math. Hung. 31 (1996), 15–25.MathSciNetzbMATHGoogle Scholar
  38. [Be87]
    W.J. Beyn. The effect of discretization on homoclinic orbits. In Bifurcation: Analysis, Algorithms, Applications 1–8, T. Küpper et al., (eds.). Birkhäuser Verlag, Basel, 1987.Google Scholar
  39. [Be90]
    W.-J. Beyn. The numerical computation of connecting orbits in dynamical systems. IMA Z. Numer. Anal, 9 (1990), 379–405.MathSciNetCrossRefGoogle Scholar
  40. [Bia196]
    V. A. Biktashev, A. V. Holden, and E. V. Nikolaev. Spiral wave meander and symmetry of the plane. Preprint, University of Leeds, 1996.Google Scholar
  41. [BiRo62]
    G. Birkhoff and G.-C. Rota. Ordinary differential equations. Ginn and Company, Boston, 1962.zbMATHGoogle Scholar
  42. [B1a100]
    P. Blancheau, J. Boissonade, and P. De Kepper. Theoretical and experimental studies of bistability in the chloride-dioxide-iodide reaction. Physica D 147 (2000), 283–299.CrossRefGoogle Scholar
  43. [Bo81a]
    R. Bogdanov. Bifurcation of the limit cycle of a family of plane vector fields. Sel. Mat. Soy. 1 (1981), 373–387.zbMATHGoogle Scholar
  44. [Bo81b]
    R. Bogdanov. Versal deformations of a singularity of a vector field on the plane in the case of zero eigenvalues. Sel. Mat. Soy., 1 (1981), 389–421.Google Scholar
  45. [BrEn93]
    M. Braune and H. Engel. Compound rotation of spiral waves in a lightsensitive Belousov-Zhabotinsky medium. Chem. Phys. Lett. 204 (1993), 257–264.CrossRefGoogle Scholar
  46. [Br64]
    R. J. Briggs. Electron-Steam Interaction With Plasmas. MIT press, Cambridge, 1964.Google Scholar
  47. [BrtD85]
    T. Bröcker and T. tom Dieck. Representations of Compact Lie Groups. Springer-Verlag, Berlin, 1985.zbMATHCrossRefGoogle Scholar
  48. [Bra101]
    H.W. Broer, B. Krauskopf, and G. Vegter. Global Analysis of Dynamical Systems. IOP Publishing, Bristol, 2001.zbMATHCrossRefGoogle Scholar
  49. [BrTa02]
    H. Broer and T. Takens (eds.). Handbook of Dynamical Systems 3. Elsevier, Amsterdam, in preparation 2002.Google Scholar
  50. [Br90]
    P. Brunovskÿ. The attracor of the scalar reaction diffusion equation is a smooth graph. J. Dynamics and Differential Equations, 2 (1990), 293–323.CrossRefGoogle Scholar
  51. [BrCh84]
    P. Brunovskÿ and S-N Chow. Generic properties of stationary state solutions of reaction-diffusion equations. J. Diff. Eqns. 53 (1984), 1–23.zbMATHCrossRefGoogle Scholar
  52. [BrFi86]
    P. Brunovskÿ and B. Fiedler. Numbers of zeros on invariant manifolds in reaction-diffusion equations. Nonlin. Analysis, TMA, 10 (1986), 179–194.zbMATHGoogle Scholar
  53. [BrFi88]
    P. Brunovskÿ and B. Fiedler. Connecting orbits in scalar reaction diffusion equations. Dynamics Reported 1 (1988), 57–89.Google Scholar
  54. [BrFi89]
    P. Brunovskÿ and B. Fiedler. Connecting orbits in scalar reaction diffusion equations II: The complete solution. J. Diff. Eqns. 81 (1989), 106–135.zbMATHCrossRefGoogle Scholar
  55. [Bra192]
    P. Brunovskÿ, P. Polâcik, and B. Sandstede. Convergence in general parabolic equations in one space dimension. Nonl. Analysis TMA 18 (1992), 209–215.zbMATHCrossRefGoogle Scholar
  56. [Caa193]
    A. Calsina, X. Mora and J. Solà-Morales. The dynamical approach to elliptic problems in cylindrical domains and a study of their parabolic singular limit. J. Diff. Eqns. 102 (1993), 244–304.zbMATHCrossRefGoogle Scholar
  57. [CaPe90]
    J. Carr and R. Pego.Invariant manifolds for metastable patterns in ust e 2 usxxf(u). Proc. Roy. Soc. Edinburgh A 116 (1990), 133–160.Google Scholar
  58. [Caa190]
    V. Castets, E. Dulos, J. Boissonade, and P. De Kepper. Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64 (1990), 2953–2956.CrossRefGoogle Scholar
  59. [ChIn74]
    N. Chafee and E. Infante. A bifurcation problem for a nonlinear parabolic equation. J. Applicable Analysis 4 (1974). 17–37.MathSciNetzbMATHCrossRefGoogle Scholar
  60. [Ch98]
    X.-Y. Chen. A strong unique continuation theorem for parabolic equations. Math. Ann. 311 (1998), 603–630.MathSciNetzbMATHCrossRefGoogle Scholar
  61. [ChVi02]
    V.V. Chepyzhov and M.I. Vishik. Attractors for Equations of Mathematical Physics. Colloq. AMS, Providence, 2002.Google Scholar
  62. [ChLa00]
    P. Chossat and R. Lauterbach. Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific, Singapore, 2000.zbMATHCrossRefGoogle Scholar
  63. [ChHa82]
    S.-N. Chow and J. K. Hale. Methods of Bifurcation Theory. Springer-Verlag, New York, 1982.zbMATHCrossRefGoogle Scholar
  64. [CoEc00]
    P. Collet and J.-P. Eckmann. Proof of the marginal stability bound for the Swift-Hohenberg equation and related equations. Preprint, 2000.Google Scholar
  65. [Co68]
    W.A. Coppel. Dichotomies and reducibility II. J. Diff. Eqns. 4 (1968), 386–398.MathSciNetzbMATHCrossRefGoogle Scholar
  66. [Co78]
    W.A. Coppel. Dichotomies in Stability Theory. Lect. Notes Math. 629, Springer, Berlin, 1978.Google Scholar
  67. [CrHo93]
    M.C. Cross and P.C. Hohenberg. Pattern formation outside equilibrium. Rev. Modern Phys. 65 (1993), 851–1112.CrossRefGoogle Scholar
  68. [Dam97]
    J. Damon. Generic properties of solutions to partial differential equations. Arch. Rat. Mech. Analysis, 140 (1997), 353–403.MathSciNetzbMATHCrossRefGoogle Scholar
  69. [DaPo02]
    E.N. Dancer and P. Polâcik. Realization of vector fields and dynamics of spatially homogeneous parabolic equations. Mein. AMS, Providence, 2002, to appear.Google Scholar
  70. [Daa196]
    G. Dangelmayr, B. Fiedler, K. Kirchgässner, and A. Mielke. Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability. Pitman 352, Boston, 1996.Google Scholar
  71. [Dea195]
    M. Dellnitz, M. Golubitsky, A. Hohmann,and I. Stewart. Spirals in scalar reaction-diffusion equations. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), 1487–1501.MathSciNetzbMATHCrossRefGoogle Scholar
  72. [De85]
    K. Deimling. Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1985.zbMATHCrossRefGoogle Scholar
  73. [dK a194]
    P. de Kepper, J.-J. Perraud, B. Rudovics,and E. Dulos. Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system. Int. J. Bifurcation Chaos Appl. Sci. Eng. 4 (1994), 1215–1231.zbMATHCrossRefGoogle Scholar
  74. [Dia195]
    O. Diekmann, S.A. v. Gils, S.M. Verduyn Lund, and H.-O. Walther. Delay Equations. Functional-, Complex-, and Nonlinear Analysis. Springer-Verlag, New York, 1995.Google Scholar
  75. [DoFr89]
    E. J. Doedel and M. J. Friedman. Numerical computation of heteroclinic orbits. J. Comp. Appl. Math. 26 (1989), 155–170.Google Scholar
  76. [DoFr91]
    E. J. Doedel and M. J. Friedman. Numerical computation and continuation of invariant manifolds connecting fixed points. SIAM J. Numer. Anal. 28 (1991), 789–808.MathSciNetzbMATHCrossRefGoogle Scholar
  77. [Doa197]
    M. Dowle, M. Mantel, and D. Barkley. Fast simulations of waves in three-dimensional excitable media. Int. J. Bifur. Chaos, 7 (1997), 2529–2546.MathSciNetzbMATHCrossRefGoogle Scholar
  78. [Dua185]
    B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov. Modern Geometry -Methods and Applications. Part 2: The Geometry and Topology of Manifolds. Springer-Verlag, New York, 1985.zbMATHCrossRefGoogle Scholar
  79. [EiYa98]
    S.-I. Ei and E. Yanagida. Slow dynamics of interfaces in the allen-cahn equation on a strip-like domain. SIAM J. Math. Anal., 29 (1998), 555–595.MathSciNetzbMATHCrossRefGoogle Scholar
  80. [Ela187]
    C. Elphick, E. Tirapegui, M.E. Brachet, P. Coullet, and G. Moss. A simple global characerization for normal forms of singular vector fields. Physica 29D (1987), 95–127.MathSciNetzbMATHGoogle Scholar
  81. [Fe71]
    N. Fenichel. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21 (1971), 193–226.MathSciNetzbMATHCrossRefGoogle Scholar
  82. [Fe74]
    N. Fenichel. Asymptotic stability with rate conditions. Indiana Univ. Math. J. 23 (1974), 1109–1137.MathSciNetzbMATHCrossRefGoogle Scholar
  83. [Fe77]
    N. Fenichel. Asymptotic stability with rate conditions, II. Indiana Univ. Math. J. 26 (1977), 81–93.MathSciNetzbMATHCrossRefGoogle Scholar
  84. [Fe79]
    N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eqns., 31 (1979), 53–98.MathSciNetzbMATHCrossRefGoogle Scholar
  85. [Fi88]
    B. Fiedler. Global Bifurcation of Periodic Solutions with Symmetry. Springer-Verlag, Berlin, 1988.zbMATHGoogle Scholar
  86. [Fi89]
    B. Fiedler. Discrete Ljapunov functionals and w-limit sets. Math. Mod. Num. Analysis, 23 (1989), 415–431.MathSciNetzbMATHGoogle Scholar
  87. [Fi94]
    B. Fiedler. Global attractors of one-dimensional parabolic equations: sixteen examples. Tatra Mountains Math. Publ., 4 (1994), 67–92.MathSciNetzbMATHGoogle Scholar
  88. [Fi96]
    B. Fiedler. Do global attractors depend on boundary conditions? Doc. Math. 1 (1996), 215–228.MathSciNetzbMATHGoogle Scholar
  89. [Fi02]
    B. Fiedler (ed.) Handbook of Dynamical Systems 2, Elsevier, Amsterdam. In press.Google Scholar
  90. FiGe98] B. Fiedler and T. Gedeon. A class of convergent neural network dynamics. Physica D,111 (1998), 288–294,.Google Scholar
  91. [FiGe99]
    B. Fiedler and T. Gedeon. A Lyapunov function for tridiagonal competitive-cooperative systems. SIAM J. Math Analysis 30 (1999), 469–478.MathSciNetzbMATHCrossRefGoogle Scholar
  92. [Fia100]
    B. Fiedler, K. Gröger, and J. Sprekels (eds.). Equadiff 99. International Conference on Differential Equations, Berlin 1999. Vol.1,2. World Scientific, Singapore, 2000.Google Scholar
  93. [FiMP89a]
    B. Fiedler and J. Mallet-Paret. Connections between Morse sets for delay-differential equations. J. reine angew. Math., 397: 23–41, (1989).MathSciNetzbMATHGoogle Scholar
  94. [FiMP89b]
    B. Fiedler and J. Mallet-Paret. A Poincaré-Bendixson theorem for scalar reaction diffusion equations. Arch. Rat. Mech. Analysis 107 (1989), 325–345.MathSciNetzbMATHCrossRefGoogle Scholar
  95. [FiMa00]
    B. Fiedler and R.-M. Mantel. Crossover collision of core filaments in three-dimensional scroll wave patterns. Doc. Math. 5 (2000), 695–731.MathSciNetzbMATHGoogle Scholar
  96. [FiPo90]
    B. Fiedler and P. Polâcik. Complicated dynamics of scalar reaction diffusion equations with a nonlocal term. Proc. Royal Soc. Edinburgh 115A (1990), 167–192.zbMATHCrossRefGoogle Scholar
  97. [FiRo96]
    B. Fiedler and C. Rocha. Heteroclinic orbits of semilinear parabolic equations. J. Diff. Eq. 125 (1996), 239–281.MathSciNetzbMATHCrossRefGoogle Scholar
  98. [FiRo99]
    B. Fiedler and C. Rocha. Realization of meander permutations by boundary value problems. J. Diff. Eqns. 156 (1999), 282–308.MathSciNetzbMATHCrossRefGoogle Scholar
  99. [FiRo00]
    B. Fiedler and C. Rocha. Orbit equivalence of global attractors of semi-linear parabolic differential equations. Trans. Amer. Math. Soc., 352 (2000), 257–284.MathSciNetzbMATHCrossRefGoogle Scholar
  100. [Fia102a]
    B. Fiedler, C. Rocha, D. Salazar, and J. Sol-Morales. A note on the dynamics of piecewise-autonomous bistable parabolic equations. Comm. Fields Inst. (2002), in press.Google Scholar
  101. [Fia102b]
    B. Fiedler, C. Rocha, and M. Wolfrum. Heteroclinic connections of S1equivariant parabolic equations on the circle. In preparation, 2002.Google Scholar
  102. [Fia196]
    B. Fiedler, B. Sandstede, A. Scheel, and C. Wulff. Bifurcation from relative equilibria of noncompact group actions: skew products, meanders and drifts. Doc. Math. J. DMV 1(1996), 479–505. See also http://www. mathematik. uni-bielef eld. de/documenta/vol-01 /20. ps. gz Google Scholar
  103. [Fia198]
    B. Fiedler, A. Scheel, and M. Vishik. Large patterns of elliptic systems in infinite cylinders. J. Math. Pures Appl. 77 (1998), 879–907.MathSciNetzbMATHGoogle Scholar
  104. [FiTu98]
    B. Fiedler and D. Turaev. Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions. Arch. Rat. Mech. Anal. 145 (1998), 129–159.MathSciNetzbMATHCrossRefGoogle Scholar
  105. [FiVi01a]
    B. Fiedler and M. Vishik. Quantitative homogenization of analytic semi-groups and reaction diffusion equations with diophantine spatial frequencies. Adv. in Diff. Eqns. 6 (2001), 1377–1408.Google Scholar
  106. [FiViOlb]
    B. Fiedler and M. Vishik. Quantitative homogenization of global at-tractors for reaction-diffusion systems with rapidly oscillating terms. Preprint, 2001.Google Scholar
  107. [Fí88]
    P.C. Fife. Dynamics of internal layers and diffusive interfaces, CBMS-NSF Reg. Conf. Ser. Appl. Math. 53, 1988.Google Scholar
  108. [Fi84]
    G. Fischer. Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen. Math. Nachr. 115 (1984), 137–157.MathSciNetzbMATHCrossRefGoogle Scholar
  109. [Fr64]
    A. Friedman. Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, New Jersey, 1964.Google Scholar
  110. [FuHa89]
    G. Fusco and J.K. Hale. Slow-motion manifolds, dormant instability, and singular perturbations. J. Dyn. Diff. Eqns. 1 (1989), 75–94.MathSciNetzbMATHCrossRefGoogle Scholar
  111. [FuO188]
    G. Fusco and W.M. Oliva. Jacobi matrices and transversality. Proc. Royal Soc. Edinburgh A 109 (1988), 231–243.MathSciNetzbMATHCrossRefGoogle Scholar
  112. [FuRo91]
    G. Fusco and C. Rocha. A permutation related to the dynamics of a scalar parabolic PDE. J. Diff. Eqns. 91 (1991), 75–94.MathSciNetCrossRefGoogle Scholar
  113. [GaHa86]
    M. Gage and R.S. Hamilton. The heat equation shrinking convex plane curves. J. Diff. Geom. 23 (1986), 69–96.MathSciNetzbMATHGoogle Scholar
  114. [GaS102]
    T. Gallay and S. Slijepcevic. Personal communication, (2002).Google Scholar
  115. [GiHi96a]
    M. Giaquinta and S. Hildebrandt. Calculus of Variations 1. The Lagrangian Formalism. Springer-Verlag, Berlin, 1996.Google Scholar
  116. [GiHi96b]
    M. Giaquinta and S. Hildebrandt. Calculus of Variations 2. The Hamiltonian Formalism. Springer-Verlag, Berlin, 1996.Google Scholar
  117. [Goa100]
    M. Golubitsky, E. Knobloch, and I. Stewart. Target patterns and spirals in planar reaction-diffusion systems. J. Nonlinear Sci. 10 (2000), 333–354.MathSciNetzbMATHCrossRefGoogle Scholar
  118. [Goa197]
    M. Golubitsky, V. LeBlanc, and I. Melbourne. Meandering of the spiral tip: an alternative approach. J. Nonl. Sci. 7 (1997), 557–586.MathSciNetzbMATHCrossRefGoogle Scholar
  119. [GoSc85]
    M. Golubitsky and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory I. Springer-Verlag, 1985.Google Scholar
  120. [Goa188]
    M. Golubitsky, I. Stewart, and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory II. Springer-Verlag, New York, 1988.zbMATHCrossRefGoogle Scholar
  121. [Goa198]
    A. Goryachev, H. Chaté, and R. Kapral. Synchronization defects and broken symmetry in spiral waves. Phys. Rev. Lett. 80 (1998), 873–876.CrossRefGoogle Scholar
  122. [Gr89]
    M. A. Grayson. Shortening embedded curves. Ann. Math. 129 (1989), 71–111.MathSciNetzbMATHCrossRefGoogle Scholar
  123. [GuHo83]
    J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983.Google Scholar
  124. [Haal]
    G. Haas, M. Bär, and I.G. Kevrekidis et al. Observation of front bifurcations in controlled geometries: From one to two dimensions. Phys. Rev. Lett. 75 (1995), 3560–3563.CrossRefGoogle Scholar
  125. [Ha82]
    P.S. Hagan. Spiral waves in reaction-diffusion equations. SIAM J. Appl. Math. 42 (1982), 762–786.MathSciNetzbMATHGoogle Scholar
  126. [Ha69]
    J.K. Hale. Ordinary Differential Equations. John Wiley Sons, New York, 1969.zbMATHGoogle Scholar
  127. [Ha85]
    J.K. Hale. Flows on centre manifolds for scalar functional differential equations. Proc. R. Soc. Edinb., Sect. A 101 (1985), 193–201.MathSciNetzbMATHCrossRefGoogle Scholar
  128. [Ha88]
    J.K. Hale. Asymptotic Behavior of Dissipative Systems. Math. Surv. 25. AMS Publications, Providence, 1988.Google Scholar
  129. [HaRa92]
    J.K. Hale and G. Raugel. Reaction-diffusion equation on thin domains. J. Math. Pures Appl. 71 (1992), 33–95.MathSciNetzbMATHGoogle Scholar
  130. [H?97]
    J. Härterich. Attractors of Viscous Balance Laws. Dissertation, Freie Universität Berlin, 1997.Google Scholar
  131. [H?98]
    J. Härterich. Attractors of viscous balance laws: Uniform estimates for the dimension. J. Diff. Eqns. 142 (1998), 188–211.zbMATHCrossRefGoogle Scholar
  132. [H?99]
    J. Härterich. Equilibrium solutions of viscous scalar balance laws with a convex flux. Nonlin. Diff. Eqns. Appl. 6 (1999), 413–436.zbMATHCrossRefGoogle Scholar
  133. [HaMi91]
    H. Hattori and K. Mischaikow. A dynamical system approach to a phase transition problem. J. Diff. Eqns. 94 (1991), 340–378.MathSciNetzbMATHCrossRefGoogle Scholar
  134. [He89]
    S. Heinze. Travelling waves for semilinear parabolic partial differential equations in cylindrical domains. Dissertation, Heidelberg, 1989.Google Scholar
  135. [He81]
    D. Henry. Geometric Theory of Semilinear Parabolic Equations. Lect. Notes Math. 804, Springer-Verlag, New York, Berlin, Heidelberg, 1981.Google Scholar
  136. [He85]
    D. Henry. Some infinite dimensional Morse-Smale systems defined by parabolic differential equations. J. Diff. Eqns. 59 (1985), 165–205.MathSciNetzbMATHCrossRefGoogle Scholar
  137. [HeWi91]
    C. Henze and A. T. Winfree. A stable knotted singularity in an excitable medium. Int. J. Bif. Chaos 1 (1991), 891–922.zbMATHCrossRefGoogle Scholar
  138. [Hi83]
    M. W. Hirsch. Differential equations and convergence almost everywhere in strongly monotone semiflows. J. Smoller, (ed.). In Nonlinear Partial Differential Equations. p. 267–285, AMS Publications, Providence, 1983.Google Scholar
  139. [Hí85]
    M. W. Hirsch. Systems of differential equations that are competitive or cooperative II. Convergence almost everywhere. SIAM J. Math. Analysis 16 (1985), 423–439.Google Scholar
  140. [Hí88]
    M. W. Hirsch. Stability and convergence in strongly monotone dynamical systems. Crelle J. reine angew. Math. 383 (1988), 1–58.zbMATHGoogle Scholar
  141. [Hia177]
    M. W. Hirsch, C.C. Pugh, and M. Shub. Invariant Manifolds. Springer-Verlag, Berlin, 1977.zbMATHGoogle Scholar
  142. [IoMi91]
    G. Moss and A. Mielke. Bifurcating time—periodic solutions of Navier-Stokes equations in infinite cylinders. J. Nonlinear Science 1 (1991), 107–146.Google Scholar
  143. [JäLu92]
    W. Jäger and S. Luckhaus. On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc., 329 (1992), 819–824.zbMATHGoogle Scholar
  144. [Jaa188]
    W. Jahnke, C. Henze, and A.T. Winfree. Chemical vortex dynamics in the 3-dimensional excitable media. Nature 336 (1988), 662–665.Google Scholar
  145. [Jaa189]
    W. Jahnke, W.E. Skaggs, and A.T. Winfree. Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable Oregonator model. J. Chem. Phys. 93 (1989), 740–749.CrossRefGoogle Scholar
  146. [Ka66]
    T. Kato. Perturbation Theory for Linear Operators. Springer, Berlin, Heidelberg, New York, 1966.Google Scholar
  147. [KaHa95]
    A. Katok and B. Hasselblatt. Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, Cambridge, 1995.Google Scholar
  148. [KaHa02]
    A. Katok and B. Hasselblatt (eds.) Handbook of Dynamical Systems 1, Elsevier, Amsterdam. to appear 2002.zbMATHGoogle Scholar
  149. [Ka87] L.H. Kauffman. On Knots. Princeton University Press, New Jersey, 1987. [Ke92]
    J.P. Keener. The core of the spiral. SIAM J. Appl. Math. 52 (1992), 1370–1390.MathSciNetzbMATHCrossRefGoogle Scholar
  150. [KeTy92]
    J.P. Keener and J.J. Tyson. The dynamics of scroll waves in excitable media. SIAM Rev., 34 (1992), 1–39.MathSciNetzbMATHCrossRefGoogle Scholar
  151. [Ki82]
    K. Kirchgässner. Wave-solutions of reversible systems and applications. J. Differential Equations 45 (1982), 113–127.MathSciNetzbMATHCrossRefGoogle Scholar
  152. [Kí00]
    S.V. Kiyashko. The generation of stable waves in faraday experiment. 2000 Int. Symp. Nonlinear Theory and its Applications, 2000.Google Scholar
  153. [KoHo73]
    N. Kopell and L.N. Howard. Plane wave solutions to reaction-diffusion equations. Studies in Appl. Math. 52 (1973), 291–328.MathSciNetzbMATHGoogle Scholar
  154. [KoHo81]
    N. Kopell and L.N. Howard. Target patterns and spiral solutions to reaction-diffusion equations with more than one space dimension. Adv. Appl. Math. 2 (1981), 417–449.MathSciNetzbMATHGoogle Scholar
  155. [Ko02]
    V.V. Kozlov. General Theory of Vortices. Enc. Math. Sc. 67, Dynamical Systems X. Springer-Verlag, New York, 2002.Google Scholar
  156. [Kr90]
    M. Krupa. Bifurcations of relative equilibria. SIAM J. Math. Analysis 21 (1990), 1453–1486.MathSciNetzbMATHCrossRefGoogle Scholar
  157. [KuMa83]
    M. Kubicek and M. Marek. Computational Methods in Bifurcation Theory and Dissipative Structures. Springer-Verlag, New York, 1983.zbMATHCrossRefGoogle Scholar
  158. [Ku95]
    Y.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer-Verlag, Berlin, 1995.zbMATHCrossRefGoogle Scholar
  159. [La91]
    O.A. Ladyzhenskaya. Attractors for Semigroups and Evolution Equations. Cambridge University Press, 1991.Google Scholar
  160. [LaLi59]
    L.D. Landau and E.M. Lifschitz. Fluid Mechanics. Pergamon Press, London, 1959.Google Scholar
  161. [Li90]
    X.-B. Lin. Using Melnikov’s method to solve Shilnikov’s problems. Proc. Roy. Soc. Edinburgh, 116A (1990), 295–325.zbMATHCrossRefGoogle Scholar
  162. [Oual]
    G. Li, Q. Ouyang, V. Petrov, and H. L. Swinney. Transition from simple rotating chemical spirals to meandering and traveling spirals. Phys. Rev. Lett. 77 (1996), 2105–2108.CrossRefGoogle Scholar
  163. [Lo04]
    A.E.H. Love. A Treatise on the Mathematical Theory of Elasticity. Dover Publications, New-York, 1904.Google Scholar
  164. [MP88]
    J. Mallet-Paret. Morse decompositions for delay-differential equations. J. Diff. Eqns. 72 (1988), 270–315.MathSciNetzbMATHCrossRefGoogle Scholar
  165. [MPSm90]
    J. Mallet-Paret and H. Smith. The Poincaré-Bendixson theorem for monotone cyclic feedback systems. J. Diff. Eqns. 4 (1990), 367–421.MathSciNetGoogle Scholar
  166. [MPSe96a]
    J. Mallet-Paret and G.R. Sell. The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay. J. Diff. Eqns. 125 (1996), 441–489.MathSciNetzbMATHCrossRefGoogle Scholar
  167. [MPSe96b]
    J. Mallet-Paret and G.R. Sell. Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions. J. Diff. Eqns. 125 (1996), 385–440.MathSciNetzbMATHCrossRefGoogle Scholar
  168. [MaPa97]
    A.F.M. Maree and A.V. Panfilov. Spiral breakup in excitable tissue due to lateral instability. Phys. Rev. Lett. 78 (1997), 1819–1822.CrossRefGoogle Scholar
  169. [Ma78]
    H. Matano. Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ., 18 (1978), 221–227.MathSciNetzbMATHGoogle Scholar
  170. [Ma79]
    H. Matano. Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. Res. Inst. Math. Sc. Kyoto Univ. 15 (1979), 401–454.MathSciNetzbMATHCrossRefGoogle Scholar
  171. [Ma82]
    H. Matano. Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation. J. Fac. Sci. Univ. Tokyo Sec. IA, 29 (1982), 401–441.MathSciNetzbMATHGoogle Scholar
  172. [Ma86]
    H. Matano. Strongly order-preserving local semi-dynamical systems the- ory and applications. In Semigroups, Theory and Applications. H. Brezis, M.G. Crandall, F. Kappel (eds.), 178–189. John Wiley Sons, New York, 1986.Google Scholar
  173. [Ma87]
    H. Matano. Strong comparison principle in nonlinear parabolic equations. In Nonlinear Parabolic Equations: Qualitative Properties of Solutions, L. Bo-cardo, A. Tesei (eds.), 148–155. Pitman Res. Notes Math. Ser. 149 (1987).Google Scholar
  174. [Ma88]
    H. Matano. Asymptotic behavior of solutions of semilinear heat equations on S’. In Nonlinear Diffusion Equations and their Equilibrium States II. W.-M. Ni, L.A. Peletier, J. Serrin (eds.). 139–162. Springer-Verlag, New York, 1988.Google Scholar
  175. [MaNa97]
    H. Matano and K.-I. Nakamura. The global attractor of semilinear parabolic equations on S l . Discr. Contin. Dyn. Syst. 3 (1997), 1–24.MathSciNetzbMATHGoogle Scholar
  176. [MaWi89]
    J. Mawhin and M. Willem. Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York, 1989.zbMATHCrossRefGoogle Scholar
  177. [Mi91]
    A. Mielke. Hamiltonian and Lagrangian Flows on Center Manifolds. Springer-Verlag, Berlin, 1991.zbMATHGoogle Scholar
  178. [Mi94]
    A. Mielke. Essential manifolds for elliptic problems in infinite cylinders. J. Diff. Eqns., 110 (1994), 322–355.MathSciNetzbMATHCrossRefGoogle Scholar
  179. [Mi97]
    A. Mielke. Instability and stability of rolls in the Swift-Hohenberg equation. Comm. Math. Phys. 189 (1997), 829–853.MathSciNetzbMATHCrossRefGoogle Scholar
  180. [MiSc96]
    A. Mielke and G. Schneider. Derivation and justification of the complex Ginzburg-Landau equation as a modulation equation. Dynamical systems and probabilistic methods in partial differential equations (Berkeley, CA, 1994 ), 191–216, Lectures in Appl. Math. 31, Amer. Math. Soc., Providence, RI, 1996.Google Scholar
  181. [MiZy91]
    A.S. Mikhailov and V.S. Zykov. Kinematical theory of spiral waves in excitable media: comparison with numerical simulations. Physica D 52 (1991), 379–397.CrossRefGoogle Scholar
  182. [Mo73]
    J. Moser. Stable and Random Motions in Dynamical Systems. Princeton University Press, New York, 1973.zbMATHGoogle Scholar
  183. [MüZy94]
    S.C. Müller and V.S. Zykov. Simple and complex spiral wave dynamics. Phil. Trans. Roy. Soc. Lond. A 347 (1994), 677–685.zbMATHCrossRefGoogle Scholar
  184. [Na90]
    N.S. Nadirashvili. On the dynamics of nonlinear parabolic equations. Soviet Math. Dokl. 40 (1990), 636–639.MathSciNetzbMATHGoogle Scholar
  185. [Neal]
    S. Nettesheim, A. von Oertzen, H.H. Rotermund, and G. Ertl. Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110) front propagation and spiral waves. J. Chem. Phys. 98 (1993), 9977–9985.CrossRefGoogle Scholar
  186. [OgNa00]
    T. Ogiwara and K.-I. Nakamura. Spiral traveling wave solutions of some parabolic equations on annuli. In Nonlinear Analysis, Josai Math. Monogr., Nishizawa, Kiyoko (ed.) 2 (2000), 15–34.Google Scholar
  187. [0102]
    W.M. Oliva. Morse-Smale semiflows. Openess and A-stability. Comm. Fields Inst. (2002), in press.Google Scholar
  188. [Pa61]
    R. S. Palais. On the existence of slices for actions of non-compact Lie groups. Ann. of Math. 73 (1961), 295–323.MathSciNetzbMATHCrossRefGoogle Scholar
  189. [Pa69]
    J. Palis. On Morse-Smale dynamical systems. Topology 8 (1969), 385–404.MathSciNetzbMATHCrossRefGoogle Scholar
  190. [PaSm70]
    J. Palis and S. Smale. Structural stability theorems. In Global Analysis. Proc. Symp. in Pure Math. vol. XIV. AMS, Providence, 1970. S. Chern, S. Smale (eds.).Google Scholar
  191. [Pa88]
    K.J. Palmer. Exponential dichotomies and Fredholm operators. Proc. Amer. Math. Soc. 104 (1988), 149–156.MathSciNetzbMATHCrossRefGoogle Scholar
  192. [PaWi85]
    A.V. Panfilov and A. T. Winfree. Dynamical simulations of twisted scroll rings in 3-dimensional excitable media. Physica D 17 (1985), 323–330.MathSciNetCrossRefGoogle Scholar
  193. [Pa83]
    A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983.zbMATHCrossRefGoogle Scholar
  194. [Pea191]
    V. Perez-Munuzuri, R. Aliev, B. Vasiev, V. Perez-Villar, and V. I. Krinsky. Super-spiral structure in an excitable medium. Nature 353 (1991) 740–742.CrossRefGoogle Scholar
  195. [Pea193]
    V. Perez-Munuzuri, M. Gomez-Gesteira, and V. Perez-Villar. A geometrical-kinematical approach to spiral wave formation: Super-spiral waves. Physica D 64 (1993), 420–430.zbMATHCrossRefGoogle Scholar
  196. [Pea197]
    D. Peterhof, A. Scheel, and B. Sandstede. Exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders. J. Diff. Eqns. 140 (1997), 266–308.MathSciNetzbMATHCrossRefGoogle Scholar
  197. [P1Bo96]
    B.B. Plapp and E. Bodenschatz. Core dynamics of multiarmed spirals in Rayleigh-Bénard convection. Physica Scripta 67 (1996), 111–117.CrossRefGoogle Scholar
  198. [Po89]
    P. Polâcik. Convergence in strongly monotone flows defined by semilinear parabolic equations. J. Diff. Eqs. 79 (1989), 89–110.zbMATHCrossRefGoogle Scholar
  199. [Po95]
    P. Polâcik. High-dimensional w-limit sets and chaos in scalar parabolic equations. J. Diff. Eqns., 119 (1995), 24–53.zbMATHCrossRefGoogle Scholar
  200. [P0021.
    P. Polâcik. Parabolic equations: Asymptotic behavior and dynamics on invariant manifolds. In Handbook of Dynamical Systems, Vol. 2. B. Fiedler (ed.), Elsevier, Amsterdam, 2002. In press.Google Scholar
  201. [Po33]
    G. Polya. Qualitatives über Wärmeaustausch. Z. Angew. Math. Mech. 13 (1933), 125–128,.Google Scholar
  202. [Po92]
    G. Pospiech. Eigenschaften, Existenz und Stabilität von travelling wave Lösungen zu einem System von Reaktions-Diffusions-Gleichungen. Dissertation, Universität Heidelberg, 1992.Google Scholar
  203. [PrRy98a]
    M. Prizzi and K.P. Rybakowski. Complicated dynamics of parabolic equations with simple gradient dependence. Trans. Am. Math. Soc. 350 (1998), 3119–3130.MathSciNetzbMATHCrossRefGoogle Scholar
  204. [PrRy98b]
    M. Prizzi and K.P. Rybakowski. Inverse problems and chaotic dynamics of parabolic equations on arbitrary spatial domains. J. Diff. Eqns. 142 (1998), 17–53.MathSciNetzbMATHCrossRefGoogle Scholar
  205. [PrWe67]
    M.H. Protter and H.F. Weinberger. Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs, New Jersey, 1967.Google Scholar
  206. [Ra02]
    G. Raugel. Global attractors. In Handbook of Dynamical Systems, Vol. 2. B. Fiedler (ed.), Elsevier, Amsterdam, 2002. In press.Google Scholar
  207. [ReSi78]
    M. Reed and B. Simon. Methods of Modern Mathematical Physics IV. Academic Press, 1978.Google Scholar
  208. [RoSa95]
    J. Robbin and D. Salamon. The spectral flow and the Maslov index. Bull. London Math. Soc. 27 (1995), 1–33.MathSciNetzbMATHCrossRefGoogle Scholar
  209. [Ro85]
    C. Rocha. Generic properties of equilibria of reaction-diffusion equations with variable diffusion.Proc. R. Soc. Edinb. A 101 (1985), 45–55.MathSciNetzbMATHCrossRefGoogle Scholar
  210. [Ro91]
    C. Rocha. Properties of the attractor of a scalar parabolic PDE. J. Dyn. Differ. Equations 3 (1991), 575–591.MathSciNetzbMATHCrossRefGoogle Scholar
  211. [Sa93a]
    B. Sandstede. Verzweigungstheorie homokliner Verdopplungen. Dissertation, Universität Stuttgart, 1993.zbMATHGoogle Scholar
  212. [Sa93b]
    B. Sandstede. Asymptotic behavior of solutions of non-autonomous scalar reaction-diffusion equations. In Conf. Proceeding International Conference on Differential Equations, Barcelona 1991, C. Perello, C. Simo, and J. Sola-Morales (eds.), 888–892, World Scientific, Singapore, 1993.Google Scholar
  213. [SaFi92]
    B. Sandstede and B. Fiedler. Dynamics of periodically forced parabolic equations on the circle. Ergod. Theor. Dynam. Sys. 12 (1992), 559–571.MathSciNetzbMATHGoogle Scholar
  214. [SaSc99]
    B. Sandstede and A. Scheel. Essential instability of pulses and bifurcations to modulated travelling waves. Proc. Roy. Soc. Edinburgh. A 129 (1999), 1263–1290.MathSciNetzbMATHCrossRefGoogle Scholar
  215. [SaScOOa]
    B. Sandstede and A. Scheel.Gluing unstable fronts and backs together can produce stable pulses. Nonlinearity 13 (2000), 1465–1482.MathSciNetzbMATHCrossRefGoogle Scholar
  216. [SaScOOb]
    B. Sandstede and A. Scheel. Spectral stability of modulated travelling waves bifurcating near essential instabilities. Proc. R. Soc. Edinburgh A 130 (2000), 419–448.MathSciNetzbMATHCrossRefGoogle Scholar
  217. [SaScOOc]
    B. Sandstede and A. Scheel. Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D 145 (2000), 233–277.MathSciNetzbMATHCrossRefGoogle Scholar
  218. [SaScOld]
    B. Sandstede and A. Scheel. Absolute versus convective instability of spiral waves. Phys. Rev. E. 62 (2000), 7708–7714.MathSciNetCrossRefGoogle Scholar
  219. [SaScOla]
    B. Sandstede and A. Scheel. Super-spiral structures of meandering and drifting spiral waves. Phys. Rev. Lett. 86 (2001), 171–174.CrossRefGoogle Scholar
  220. [SaScOlb]
    B. Sandstede and A. Scheel. Essential instabilities of fronts: bifurcation and bifurcation failure. Dynamical Systems: An International Journal 16 (2001), 1–28.MathSciNetzbMATHGoogle Scholar
  221. [SaScOlc]
    B. Sandstede and A. Scheel. On the stability of periodic travelling waves with large spatial period. J. Diff. Eqns. 172 (2001), 134–188.MathSciNetzbMATHCrossRefGoogle Scholar
  222. [SaScOld]
    B. Sandstede and A. Scheel. On the structure of spectra of modulated travelling waves. Math. Nachr. 232 (2001), 39–93.MathSciNetzbMATHCrossRefGoogle Scholar
  223. [SaScO2a]
    B. Sandstede and A. Scheel. Nonlinear convective stability and instability the role of absolute spectra and nonlinearities.In preparation (2002).Google Scholar
  224. [SaScO2b]
    B. Sandstede and A. Scheel. Instabilities of spiral waves in large disks.In preparation (2002).Google Scholar
  225. [Saa197a]
    B. Sandstede, A. Scheel, and C. Wulff. Center manifold reduction for spiral wave dynamics. C. R. Acad. Sci. Paris, Série 1324 (1997), 153–158.Google Scholar
  226. [Saa197b]
    B. Sandstede, A. Scheel, and C. Wulff. Dynamics of spiral waves on unbounded domains using center-manifold reduction. J. Diff. Eqns. 141 (1997), 122–149.MathSciNetzbMATHCrossRefGoogle Scholar
  227. [Saa199]
    B. Sandstede, A. Scheel, and C. Wulff. Bifurcations and dynamics of spiral waves. J. Nonlinear Science 9 (1999), 439–478.MathSciNetzbMATHCrossRefGoogle Scholar
  228. [Sc90]
    R. Schaaf. Global Solution Branches of Two Point Boundary Value Problems. Springer-Verlag, New York, 1990.zbMATHGoogle Scholar
  229. [Sc96]
    A. Scheel. Existence of fast travelling waves for some parabolic equations —a dynamical systems approach. J. Dyn. Diff. Eqns. 8 (1996), 469–548.MathSciNetzbMATHCrossRefGoogle Scholar
  230. [Sc97]
    A. Scheel. Subcritical bifurcation to infinitely many rotating waves. J. Math. Anal. Appl. 215 (1997), 252–261.MathSciNetzbMATHCrossRefGoogle Scholar
  231. [Sc98]
    A. Scheel. Bifurcation to spiral waves in reaction-diffusion systems. SIAM J. Math. Anal. 29 (1998), 1399–1418.MathSciNetzbMATHCrossRefGoogle Scholar
  232. [Sc01]
    A. Scheel.Radially symmetric patterns of reaction-diffusion systems. Preprint 2001.Google Scholar
  233. [Sc98b]
    G. Schneider. Hopf bifurcation in spatially extended reaction-diffusion systems. J. Nonlinear Sci. 8 (1998), 17–41.MathSciNetzbMATHCrossRefGoogle Scholar
  234. [Sc98c]
    G. Schneider. Nonlinear diffusive stability of spatially periodic solutions — abstract theorem and higher space dimensions. Tohoku Math. Publ. 8 (1998), 159–167.Google Scholar
  235. [Si89]
    Ya.G. Sinai. Ergodic theory with applications to dynamical systems and statistical mechanics. Enc. Math. Sc. 2, Dynamical Systems II. Springer-Verlag, Berlin, 1989.Google Scholar
  236. [SkSw91]
    G. S. Skinner and H. L. Swinney. Periodic to quasiperiodic transition of chemical spiral rotation. Physica D 48 (1991), 1–16.zbMATHCrossRefGoogle Scholar
  237. [Sm95]
    H. Smith. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. AMS, Providence, 1995.zbMATHGoogle Scholar
  238. [Sm83]
    J. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York, 1983.zbMATHCrossRefGoogle Scholar
  239. [St00]
    A. Steven. The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math., 61 (2000), 183–212.MathSciNetCrossRefGoogle Scholar
  240. [St90]
    M. Struwe. Variational Methods. Springer-Verlag, Berlin, 1990.zbMATHCrossRefGoogle Scholar
  241. St36] C. Sturm. Sur une classe d’équations à différences partielles. J. Math. Pure Appl. 1(1836), 373–444,.Google Scholar
  242. [Ta74]
    F. Takens. Singularities of vector fields. Publ. IHES, 43 (1974), 47–100.MathSciNetGoogle Scholar
  243. [Ta79]
    H. Tanabe. Equations of Evolution. Pitman, Boston, 1979.zbMATHGoogle Scholar
  244. [Te88]
    R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York, 1988.zbMATHCrossRefGoogle Scholar
  245. [ToKn98]
    S.M. Tobias and E. Knobloch. Breakup of spiral waves into chemical turbulence. Phys. Rev. Lett. 80 (1998), 4811–4814.CrossRefGoogle Scholar
  246. [Tu52]
    A. Turing. The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. B 237 (1952), 37–72.CrossRefGoogle Scholar
  247. [TySt91]
    J.J. Tyson and S.H. Strogatz. The differential geometry of scroll waves. Int. J. Bif. Chaos, 1 (1991), 723–744.MathSciNetzbMATHCrossRefGoogle Scholar
  248. [Uh76]
    K. Uhlenbeck. Generic properties of eigenfunctions. Amer. J. Math., 98 (1976), 1059–1078.MathSciNetzbMATHCrossRefGoogle Scholar
  249. [Una193]
    Zs. Ungvarai-Nagy, J. Ungvarai, and S.C. Müller. Complexity in spiral wave dynamics. Chaos 3 (1993), 15–19.CrossRefGoogle Scholar
  250. [Va82]
    A. Vanderbauwhede. Local Bifurcation and Symmetry. Pitman, Boston, 1982.zbMATHGoogle Scholar
  251. [Va89]
    A. Vanderbauwhede. Center manifolds, normal forms and elementary bifurcations. Dynamics Reported 2 (1989), 89–169.MathSciNetCrossRefGoogle Scholar
  252. [Wa70]
    W. Walter. Differential and Integral Inequalities. Springer-Verlag, New York, 1970.zbMATHCrossRefGoogle Scholar
  253. [WiRo46]
    N. Wiener and A. Rosenblueth. The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch. Inst. Cardiol. Mexico 16 (1946), 205–265.Google Scholar
  254. [Wi72]
    A. T. Winfree. Spiral waves of chemical activity. Science, 175 (1972), 634–636.CrossRefGoogle Scholar
  255. [Wi73]
    A. T. Winfree. Scroll-shaped waves of chemical activity in three dimensions. Science 181 (1973), 937–939.Google Scholar
  256. [Wi87]
    A. T. Winfree. When Time Breaks Down. Princeton University Press, Princeton, NJ, 1987.Google Scholar
  257. [Wi91]
    A. T. Winfree. Varieties of spiral wave behavior: An experimentalist’s approach to the theory of excitable media. Chaos 1 (1991), 303–334.Google Scholar
  258. [Wi95]
    A. T. Winfree. Persistent tangles of vortex rings in excitable media. Physica D 84 (1995), 126–147.Google Scholar
  259. [WiOl]
    A. T. Winfree. The geometry of biological time. Biomathematics 8, Springer-Verlag, Berlin-New York, 2001.Google Scholar
  260. [Wia195]
    A. T. Winfree, E.M. Winfree, and M. Seifert. Organizing centers in a cellular excitable medium. Physica D, 17 (1995), 109–115.MathSciNetCrossRefGoogle Scholar
  261. [Wo98]
    Matthias Wolfrum. Geometry of Heteroclinic Cascades in Scalar Semilinear Parabolic Equations. Dissertation, Freie Universität Berlin, 1998.Google Scholar
  262. [Wo02a]
    M. Wolfrum. Personal communication, (2002).Google Scholar
  263. [Wo02b]
    M. Wolfrum. A sequence of order relations, encoding heteroclinic connections in scalar parabolic PDEs. J. Diff. Eqns., to appear (2002).Google Scholar
  264. [Wu96]
    C. Wulff. Theory of Meandering and Drifting Spiral Waves in Reaction-Diffusion Systems. Dissertation, Berlin, 1996.Google Scholar
  265. [Wua101]
    C. Wulff, J. Lamb, and I. Melbourne. Bifurcation from relative periodic solutions. Ergodic Theory Dynam. Systems 21 (2001), 605–635.Google Scholar
  266. [Yaa198]
    H. Yagisita, M. Mimura, and M. Yamada. Spiral wave behaviors in an excitable reaction-diffusion system on a sphere. Physica D 124 (1998), 126–136.Google Scholar
  267. [Ze85]
    E. Zeidler. Nonlinear functional analysis and its applications. III: Variational methods and optimization. Springer-Verlag, New York, 1985.zbMATHGoogle Scholar
  268. [Ze93]
    E. Zeidler. Nonlinear functional analysis and its applications. Volume I: Fixed-point theorems. Springer-Verlag, New York, 1993.zbMATHGoogle Scholar
  269. [Ze68]
    T.I. Zelenyak. Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable. Diff. Eqns. 4 (1968), 17–22.Google Scholar
  270. [ZhOu00]
    L. Q. Zhou and Q. Ouyang. Experimental studies on long-wavelength instability and spiral breakup in a reaction-diffusion system. Phys. Rev. Lett. 85 (2000), 1650–1653.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Bernold Fiedler
    • 1
  • Arnd Scheel
    • 2
  1. 1.FB Mathematik IFreie Universität BerlinBerlinGermany
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations