AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Spiral Waves: Linear and Nonlinear Theory
About this Title
Björn Sandstede and Arnd Scheel
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 285, Number 1413
ISBNs: 978-1-4704-6309-0 (print); 978-1-4704-7483-6 (online)
DOI: https://doi.org/10.1090/memo/1413
Published electronically: April 25, 2023
Keywords: Spiral waves,
existence,
spectrum,
stability,
asymptotic expansions
Table of Contents
Chapters
- 1. Introduction
- 2. Background Material on Wave Trains
- 3. Main Results
- 4. Wave Trains
- 5. Exponential Dichotomies
- 6. Fredholm Properties
- 7. Robustness and Asymptotics of Spiral Waves
- 8. Shape of Eigenfunctions, and Transverse Instabilities
- 9. Spiral Waves on Large Finite Disks
- 10. Spectra of Spiral Waves Restricted to Large Finite Disks
- 11. Spectra of Truncated Spiral Waves
- 12. Applications to Spiral-Wave Dynamics and Discussion
- A. Numerical Computation of Spiral Waves in Model Systems
Abstract
Spiral waves are striking self-organized coherent structures that organize spatio-temporal dynamics in dissipative, spatially extended systems. In this paper, we provide a conceptual approach to various properties of spiral waves. Rather than studying existence in a specific equation, we study properties of spiral waves in general reaction-diffusion systems. We show that many features of spiral waves are robust and to some extent independent of the specific model analyzed. To accomplish this, we present a suitable analytic framework, spatial radial dynamics, that allows us to rigorously characterize features such as the shape of spiral waves and their eigenfunctions, properties of the linearization, and finite-size effects. We believe that our framework can also be used to study spiral waves further and help analyze bifurcations, as well as provide guidance and predictions for experiments and numerical simulations. From a technical point of view, we introduce non-standard function spaces for the well-posedness of the existence problem which allow us to understand properties of spiral waves using dynamical systems techniques, in particular exponential dichotomies. Using these pointwise methods, we are able to bring tools from the analysis of one-dimensional coherent structures such as fronts and pulses to bear on these inherently two-dimensional defects.- S. Alonso, M. Bär, and B. Echebarria, Nonlinear physics of electrical wave propagation in the heart: a review, Rep. Prog. Phys. 79 (2016), no. 9, 096601–57.
- I. S. Aranson, L. Aranson, L. Kramer, and A. Weber, Stability limits of spirals and traveling waves in nonequilibrium media, Phys. Rev. A 46 (1992), R2992–R2995.
- Igor S. Aranson and Lorenz Kramer, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys. 74 (2002), no. 1, 99–143. MR 1895097, DOI 10.1103/RevModPhys.74.99
- Peter Ashwin and Ian Melbourne, Noncompact drift for relative equilibria and relative periodic orbits, Nonlinearity 10 (1997), no. 3, 595–616. MR 1448578, DOI 10.1088/0951-7715/10/3/002
- Peter Ashwin, Ian Melbourne, and Matthew Nicol, Drift bifurcations of relative equilibria and transitions of spiral waves, Nonlinearity 12 (1999), no. 4, 741–755. MR 1709854, DOI 10.1088/0951-7715/12/4/301
- M. Bär and M. Eiswirth, Turbulence due to spiral breakup in a continuous excitable medium, Phys. Rev. E 48 (1993), R1635–R1637.
- 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.
- D. Barkley, Linear stability analysis of rotating spiral waves in excitable media., Phys. Rev. Lett. 68 (1992), no. 13, 2090–2093.
- Dwight Barkley, Mark Kness, and Laurette S. Tuckerman, Spiral-wave dynamics in a simple model of excitable media: the transition from simple to compound rotation, Phys. Rev. A (3) 42 (1990), no. 4, 2489–2491. MR 1068482, DOI 10.1103/PhysRevA.42.2489
- D. Barkley, Euclidean symmetry and the dynamics of rotating spiral waves, Phys. Rev. Lett. 72 (1994), 164–167.
- D. Barkley, EZ-SPIRAL, 2007.
- Margaret Beck, Graham Cox, Christopher Jones, Yuri Latushkin, and Alim Sukhtayev, A dynamical approach to semilinear elliptic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), no. 2, 421–450. MR 4211992, DOI 10.1016/j.anihpc.2020.08.001
- M. Beck, G. Cox, C. Jones, Y. Latushkin, and A. Sukhtayev, Exponential dichotomies for elliptic pde on radial domains, Preprint, 2019, arXiv:1907.10372.
- Margaret Beck, Toan T. Nguyen, Björn Sandstede, and Kevin Zumbrun, Nonlinear stability of source defects in the complex Ginzburg-Landau equation, Nonlinearity 27 (2014), no. 4, 739–786. MR 3190319, DOI 10.1088/0951-7715/27/4/739
- Jeremy Bellay and Arnd Scheel, Coherent structures near the boundary between excitable and oscillatory media, Dyn. Syst. 25 (2010), no. 1, 111–132. MR 2765451, DOI 10.1080/14689360903325071
- A. Belmonte, O. Qi, and J. Flesselles, Experimental survey of spiral dynamics in the Belousov-Zhabotinsky reaction, Journal De Physique II 7 (1997), no. 10, 1425–1468.
- Andrew J. Bernoff, Spiral wave solutions for reaction-diffusion equations in a fast reaction/slow diffusion limit, Phys. D 53 (1991), no. 1, 125–150. MR 1137454, DOI 10.1016/0167-2789(91)90168-9
- Olivier Bernus, Arun V. Holden, and Alexander V. Panfilov, Nonlinear waves in excitable media: approaches to cardiac arrhythmias, Phys. D 238 (2009), no. 11-12, v–viii. MR 2527156
- G. Bertin and C. Lin, Spiral structure in galaxies: A density wave theory, MIT press, 1996.
- M. Bestehorn, M. Fantz, R. Friedrich, and H. Haken, Hexagonal and spiral patterns of thermal convection, Phys. Lett. A 174 (1993), no. 1, 48–52.
- V. N. Biktashev, A. V. Holden, and H. Zhang, Tension of organizing filaments of scroll waves, Philos. Trans. Roy. Soc. London Ser. A 347 (1994), no. 1685, 611–630. MR 1404227, DOI 10.1098/rsta.1994.0070
- I. V. Biktasheva, D. Barkley, V. N. Biktashev, G. V. Bordyugov, and A. J. Foulkes, Computation of the response functions of spiral waves in active media, Phys. Rev. E (3) 79 (2009), no. 5, 056702, 10. MR 2551439, DOI 10.1103/PhysRevE.79.056702
- I.V. Biktasheva, Y.E. Elkin, and V.N. Biktashev, Localized sensitivity of spiral waves in the complex Ginzburg-Landau equation, Phys. Rev. E 57 (1998), no. 3, 2656–2659.
- I. V. Biktasheva, A. V. Holden, and V. N. Biktashev, Localization of response functions of spiral waves in the FitzHugh-Nagumo system, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 16 (2006), no. 5, 1547–1555. MR 2254871, DOI 10.1142/S0218127406015490
- Grigori Bordiougov and Harald Engel, From trigger to phase waves and back again, Phys. D 215 (2006), no. 1, 25–37. MR 2232442, DOI 10.1016/j.physd.2006.01.005
- Paul Carter and Björn Sandstede, Unpeeling a homoclinic banana in the FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst. 17 (2018), no. 1, 236–349. MR 3755655, DOI 10.1137/16M1080707
- E. Cherry, F. Fenton, T. Krogh-Madsen, S. Luther, and U. Parlitz, Introduction to Focus Issue: Complex Cardiac Dynamics, Chaos 27 (2017), no. 9, 093701–9.
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 69338
- W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR 481196
- Graham Cox, Christopher K. R. T. Jones, Yuri Latushkin, and Alim Sukhtayev, The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials, Trans. Amer. Math. Soc. 368 (2016), no. 11, 8145–8207. MR 3546796, DOI 10.1090/tran/6801
- J. Davidsen, R. Erichsen, R. Kapral, and H. Chate, From ballistic to Brownian vortex motion in complex oscillatory media, Phys. Rev. Lett. 93 (2004), no. 1, 018305.
- R. Desai and R. Kapral, Dynamics of self-organized and self-assembled structures, Cambridge University Press, 2009.
- Stephanie Dodson and Björn Sandstede, Determining the source of period-doubling instabilities in spiral waves, SIAM J. Appl. Dyn. Syst. 18 (2019), no. 4, 2202–2226. MR 4041706, DOI 10.1137/19M1264813
- S. Dodson and Björn Sandstede, GitHub Repository, 2019.
- Arjen Doelman, Björn Sandstede, Arnd Scheel, and Guido Schneider, The dynamics of modulated wave trains, Mem. Amer. Math. Soc. 199 (2009), no. 934, viii+105. MR 2507940, DOI 10.1090/memo/0934
- Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
- Bernold Fiedler, Björn Sandstede, Arnd Scheel, and Claudia Wulff, Bifurcation from relative equilibria of noncompact group actions: skew products, meanders, and drifts, Doc. Math. 1 (1996), No. 20, 479–505. MR 1425301
- Bernold Fiedler and Dmitry Turaev, Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions, Arch. Ration. Mech. Anal. 145 (1998), no. 2, 129–159. MR 1664546, DOI 10.1007/s002050050126
- M. Osman Gani and T. Ogawa, Alternans and spiral breakup in an excitable reaction-diffusion system: a simulation study, Int. Sch. Res. Notices 2014 (2014), 459675.
- R. A. Gardner, On the structure of the spectra of periodic travelling waves, J. Math. Pures Appl. (9) 72 (1993), no. 5, 415–439. MR 1239098
- Robert A. Gardner and Kevin Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math. 51 (1998), no. 7, 797–855. MR 1617251, DOI 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1
- M. Golubitsky, V. G. LeBlanc, and I. Melbourne, Meandering of the spiral tip: an alternative approach, J. Nonlinear Sci. 7 (1997), no. 6, 557–586. MR 1474642, DOI 10.1007/s003329900040
- A. Goryachev, H. Chaté, and R. Kapral, Synchronization defects and broken symmetry in spiral waves, Phys. Rev. Lett. 80 (1998), 873–876.
- J. M. Greenberg, Spiral waves for $\lambda -\omega$ systems, SIAM J. Appl. Math. 39 (1980), no. 2, 301–309. MR 588502, DOI 10.1137/0139026
- J. M. Greenberg, Spiral waves for $\lambda -\omega$ systems. II, Adv. in Appl. Math. 2 (1981), no. 4, 450–454. MR 639765, DOI 10.1016/0196-8858(81)90044-0
- Patrick S. Hagan, Spiral waves in reaction-diffusion equations, SIAM J. Appl. Math. 42 (1982), no. 4, 762–786. MR 665385, DOI 10.1137/0142054
- A. Hagberg and E. Meron, Complex patterns in reaction-diffusion systems: A tale of two front instabilities, Chaos 4 (1994), no. 3, 477–484.
- Vincent Hakim and Alain Karma, Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications, Phys. Rev. E (3) 60 (1999), no. 5, 5073–5105. MR 1719409, DOI 10.1103/PhysRevE.60.5073
- Jörg Härterich, Björn Sandstede, and Arnd Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J. 51 (2002), no. 5, 1081–1109. MR 1947869, DOI 10.1512/iumj.2002.51.2188
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
- Sebastian Hermann and Georg A. Gottwald, The large core limit of spiral waves in excitable media: a numerical approach, SIAM J. Appl. Dyn. Syst. 9 (2010), no. 2, 536–567. MR 2670009, DOI 10.1137/090780055
- Peter Howard, Pointwise estimates and stability for degenerate viscous shock waves, J. Reine Angew. Math. 545 (2002), 19–65. MR 1896097, DOI 10.1515/crll.2002.034
- G. Iooss, A. Mielke, and Y. Demay, Theory of steady Ginzburg-Landau equation, in hydrodynamic stability problems, European J. Mech. B Fluids 8 (1989), no. 3, 229–268. MR 1008662
- 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. Phys. Chem. 93 (1989), no. 2, 740–749.
- Todd Kapitula and Björn Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Phys. D 124 (1998), no. 1-3, 58–103. MR 1662530, DOI 10.1016/S0167-2789(98)00172-9
- A Karma, Electrical alternans and spiral wave breakup in cardiac tissue, Chaos 4 (1994), no. 3, 461–472.
- Alain Karma, Spiral breakup in model equations of action potential propagation in cardiac tissue, Phys. Rev. Lett. 71 (1993), no. 7, 1103–1106. MR 1231536, DOI 10.1103/PhysRevLett.71.1103
- James P. Keener and John J. Tyson, The dynamics of scroll waves in excitable media, SIAM Rev. 34 (1992), no. 1, 1–39. MR 1156287, DOI 10.1137/1034001
- Klaus Kirchgässner, Wave-solutions of reversible systems and applications, J. Differential Equations 45 (1982), no. 1, 113–127. MR 662490, DOI 10.1016/0022-0396(82)90058-4
- N. Kopell and L. N. Howard, Plane wave solutions to reaction-diffusion equations, Studies in Appl. Math. 52 (1973), 291–328. MR 359550, DOI 10.1002/sapm1973524291
- N. Kopell and L. N. Howard, Target pattern and spiral solutions to reaction-diffusion equations with more than one space dimension, Adv. in Appl. Math. 2 (1981), no. 4, 417–449. MR 639764, DOI 10.1016/0196-8858(81)90043-9
- Peter Kuchment, Floquet theory for partial differential equations, Operator Theory: Advances and Applications, vol. 60, Birkhäuser Verlag, Basel, 1993. MR 1232660, DOI 10.1007/978-3-0348-8573-7
- Yuri Latushkin and Alin Pogan, The dichotomy theorem for evolution bi-families, J. Differential Equations 245 (2008), no. 8, 2267–2306. MR 2446192, DOI 10.1016/j.jde.2008.01.023
- Yuri Latushkin and Alin Pogan, The infinite dimensional Evans function, J. Funct. Anal. 268 (2015), no. 6, 1509–1586. MR 3306356, DOI 10.1016/j.jfa.2014.11.020
- 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.
- Christopher D. Marcotte and Roman O. Grigoriev, Unstable spiral waves and local Euclidean symmetry in a model of cardiac tissue, Chaos 25 (2015), no. 6, 063116, 16. MR 3369840, DOI 10.1063/1.4922596
- Christopher D. Marcotte and Roman O. Grigoriev, Adjoint eigenfunctions of temporally recurrent single-spiral solutions in a simple model of atrial fibrillation, Chaos 26 (2016), no. 9, 093107, 14. MR 3546771, DOI 10.1063/1.4962644
- A. F. M. Marée and A. V. Panfilov, Spiral breakup in excitable tissue due to lateral instability, Phys. Rev. Lett. 78 (1997), 1819–1822.
- Daniel Margerit and Dwight Barkley, Cookbook asymptotics for spiral and scroll waves in excitable media, Chaos 12 (2002), no. 3, 636–649. MR 1939456, DOI 10.1063/1.1494875
- Alexander Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci. 10 (1988), no. 1, 51–66. MR 929221, DOI 10.1002/mma.1670100105
- Alexander Mielke, Hamiltonian and Lagrangian flows on center manifolds, Lecture Notes in Mathematics, vol. 1489, Springer-Verlag, Berlin, 1991. With applications to elliptic variational problems. MR 1165943, DOI 10.1007/BFb0097544
- Gerhard Dangelmayr, Bernold Fiedler, Klaus Kirchgässner, and Alexander Mielke, Dynamics of nonlinear waves in dissipative systems: reduction, bifurcation and stability, Pitman Research Notes in Mathematics Series, vol. 352, Longman, Harlow, 1996. With a contribution by G. Raugel. MR 1458064
- A. Milke and S. Zelik, Infinite-dimensional trajectory attractors of elliptic boundary value problems in cylindrical domains, Uspekhi Mat. Nauk 57 (2002), no. 4(346), 119–150 (Russian, with Russian summary); English transl., Russian Math. Surveys 57 (2002), no. 4, 753–784. MR 1942119, DOI 10.1070/RM2002v057n04ABEH000550
- Q. Ouyang and J.-M. Flesselles, Transition from spirals to defect turbulence driven by a convective instability, Nature 379 (1996), no. 6561, 143–146.
- Frank Pacard and Tristan Rivière, Linear and nonlinear aspects of vortices, Progress in Nonlinear Differential Equations and their Applications, vol. 39, Birkhäuser Boston, Inc., Boston, MA, 2000. The Ginzburg-Landau model. MR 1763040, DOI 10.1007/978-1-4612-1386-4
- Kenneth J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), no. 2, 225–256. MR 764125, DOI 10.1016/0022-0396(84)90082-2
- Kenneth J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc. 104 (1988), no. 1, 149–156. MR 958058, DOI 10.1090/S0002-9939-1988-0958058-1
- Fan Yang and Howard A. Stone, Formation, rupture, and healing of an annular viscous film, Phys. Rev. Lett. 124 (2020), no. 22, 224501, 5. MR 4111209, DOI 10.1103/physrevlett.124.224501
- J. Pastore, S. Girouard, K.R. Laurita, F.G. Akar, and D.S. Rosenbaum, Mechanism linking T-wave alternans to the genesis of cardiac fibrillation, Circulation 99 (1999), no. 10, 1385–1394.
- D. E. Pelinovsky and A. Scheel, Stability analysis of stationary light transmission in nonlinear photonic structures, J. Nonlinear Sci. 13 (2003), no. 4, 347–396. MR 1992381, DOI 10.1007/s00332-003-0527-3
- V. Perez-Muñuzuri, R. Aliev, B. Vasiev, V. Perez-Villar, and V. I. Krinsky, Super-spiral structures in an excitable medium, Nature 353 (1991), no. 6346, 740–742.
- Daniela Peterhof, Björn Sandstede, and Arnd Scheel, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders, J. Differential Equations 140 (1997), no. 2, 266–308. MR 1483003, DOI 10.1006/jdeq.1997.3303
- L. M. Pismen, Patterns and interfaces in dissipative dynamics, Springer Series in Synergetics, Springer-Verlag, Berlin, 2006. With a foreword by Y. Pomeau. MR 2235817
- T. Plesser, S.C. Mueller, and B. Hess, Spiral wave dynamics as a function of proton concentration in the ferroin-catalyzed Belousov-Zhabotinskii reaction, J. Phys. Chem. 94 (1990), no. 19, 7501–7507.
- Jens D. M. Rademacher, Björn Sandstede, and Arnd Scheel, Computing absolute and essential spectra using continuation, Phys. D 229 (2007), no. 2, 166–183. MR 2341158, DOI 10.1016/j.physd.2007.03.016
- D.S. Rosenbaum, L.E. Jackson, J.M. Smith, H. Garan, J.N. Ruskin, and R.J. Cohen, Electrical Alternans and Vulnerability to Ventricular Arrhythmias, New England J. Medicine 330 (1994), no. 4, 235–241.
- E. Rössler and C. Kahlert, Winfree meandering in a 2-dimensional 2-variable excitable medium, Zeitschr. für Naturforschung A 34 (1979), 565–570.
- Violaine Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 21 (2004), no. 3, 341–379 (English, with English and French summaries). MR 2068306, DOI 10.1016/S0294-1449(03)00042-8
- Björn Sandstede, Convergence estimates for the numerical approximation of homoclinic solutions, IMA J. Numer. Anal. 17 (1997), no. 3, 437–462. MR 1459332, DOI 10.1093/imanum/17.3.437
- B. Sandstede, A. Scheel, and C. Wulff, Bifurcations and dynamics of spiral waves, J. Nonlinear Sci. 9 (1999), no. 4, 439–478. MR 1700672, DOI 10.1007/s003329900076
- Björn Sandstede and Arnd Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Phys. D 145 (2000), no. 3-4, 233–277. MR 1782392, DOI 10.1016/S0167-2789(00)00114-7
- Björn Sandstede and Arnd Scheel, Absolute versus convective instability of spiral waves, Phys. Rev. E (3) 62 (2000), no. 6, 7708–7714. MR 1807181, DOI 10.1103/PhysRevE.62.7708
- Björn Sandstede and Arnd Scheel, On the structure of spectra of modulated travelling waves, Math. Nachr. 232 (2001), 39–93. MR 1871473, DOI 10.1002/1522-2616(200112)232:1<39::AID-MANA39>3.3.CO;2-X
- B. Sandstede and A. Scheel, Superspiral structures of meandering and drifting spiral waves, Phys. Rev. Lett. 86 (2001), 171–174.
- Björn Sandstede and Arnd Scheel, Defects in oscillatory media: toward a classification, SIAM J. Appl. Dyn. Syst. 3 (2004), no. 1, 1–68. MR 2067899, DOI 10.1137/030600192
- B. Sandstede, and A. Scheel, Curvature effects on spiral spectra: Generation of point eigenvalues near branch points, Phys. Rev. E 73 (2006), 016217.
- Björn Sandstede and Arnd Scheel, Relative Morse indices, Fredholm indices, and group velocities, Discrete Contin. Dyn. Syst. 20 (2008), no. 1, 139–158. MR 2350063, DOI 10.3934/dcds.2008.20.139
- Björn Sandstede and Arnd Scheel, Gluing unstable fronts and backs together can produce stable pulses, Nonlinearity 13 (2000), no. 5, 1465–1482. MR 1781803, DOI 10.1088/0951-7715/13/5/303
- Björn Sandstede and Arnd Scheel, Period-doubling of spiral waves and defects, SIAM J. Appl. Dyn. Syst. 6 (2007), no. 2, 494–547. MR 2318665, DOI 10.1137/060668158
- B. Sandstede and A. Scheel, GitHub Repository, 2020.
- Björn Sandstede, Arnd Scheel, and Claudia Wulff, Dynamics of spiral waves on unbounded domains using center-manifold reductions, J. Differential Equations 141 (1997), no. 1, 122–149. MR 1485945, DOI 10.1006/jdeq.1997.3326
- Arnd Scheel, Bifurcation to spiral waves in reaction-diffusion systems, SIAM J. Math. Anal. 29 (1998), no. 6, 1399–1418. MR 1638058, DOI 10.1137/S0036141097318948
- Arnd Scheel, Radially symmetric patterns of reaction-diffusion systems, Mem. Amer. Math. Soc. 165 (2003), no. 786, viii+86. MR 1997690, DOI 10.1090/memo/0786
- Barry Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Physics 97 (1976), no. 2, 279–288. MR 404846, DOI 10.1016/0003-4916(76)90038-5
- G.S. Skinner and H.L. Swinney, Periodic to quasiperiodic transition of chemical spiral rotation, Physica D 48 (1991), no. 1, 1–16.
- John J. Tyson and James P. Keener, Singular perturbation theory of traveling waves in excitable media (a review), Phys. D 32 (1988), no. 3, 327–361. MR 980194, DOI 10.1016/0167-2789(88)90062-0
- Paul Wheeler and Dwight Barkley, Computation of spiral spectra, SIAM J. Appl. Dyn. Syst. 5 (2006), no. 1, 157–177. MR 2217134, DOI 10.1137/050624273
- Norbert Wiener and Arturo 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. México 16 (1946), 205–265 (English, with Spanish summary). MR 25140
- Arthur T. Winfree, The geometry of biological time, Biomathematics, vol. 8, Springer-Verlag, Berlin-New York, 1980. MR 572965
- Arthur T. Winfree, Varieties of spiral wave behavior: an experimentalist’s approach to the theory of excitable media, Chaos 1 (1991), no. 3, 303–334. MR 1135914, DOI 10.1063/1.165844
- M. Yoneyama, A. Fujii, and S. Maeda, Wavelength-doubled spiral fragments in photosensitive monolayers, J. Amer. Chem. Soc. 117 (1995), no. 31, 8188–8191.