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References
Angenent, S.: Parabolic equations for curves on surfaces. I: Curves with p-integrable curvature. Ann. Math. 132, 451–483 (1990)
Angenent, S.: Parabolic equations for curves on surfaces. II: Intersections, blow-up and generalized solutions. Ann. Math. 133, 171–251 (1991)
Angenent, S.: Inflection points, extatic points and curve shortening. In: Simó, C. (ed.), Hamiltonian systems with three or more degrees of freedom. Dordrecht: Kluwer. NATO ASI Ser., Ser. C, Math. Phys. Sci. 533, 3–10 (1999)
Arnol’d, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Berlin, Heidelberg, New York: Springer 1983
Braune, M., Engel, H.: Compound rotation of spiral waves in a light-sensitive Belousov-Zhabotinsky medium. Chem. Phys. Lett. 204, 257–264 (1993)
Belitskii, G.R.: Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class. Funct. Anal. Appl. 7, 268–277 (1973)
Chow, S.-N., Hale, J.K.: Methods of Bifurcation Theory. Berlin, Heidelberg, New York: Springer 1982
Fiedler, B., Guo, J.-S., Tsai, J.-C.: Multiplicity of rotating spirals under curvature flows with normal tip motion. J. Differ. Equations (in press) (2004)
Fiedler, B., Mantel, R.-M.: Crossover collision of core filaments in three-dimensional scroll wave patterns. Doc. Math. 5, 695–731 (2000)
Fiedler, B., Scheel, A.: Spatio-temporal dynamics of reaction-diffusion patterns. In: Kirkilionis, M., Kromker, S., Rannacher, R., Tomi F. (eds.), Trends in Nonlinear Analysis, pp. 23–142. Berlin, Heidelberg, New York: Springer 2003
Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)
Ikota, R., Ishimura, N., Yamaguchi, T.: On the structures of steady solutions for the kinematic model of spiral waves in excitable media. Jpn. J. Ind. Appl. Math. 15, 317–330 (1998)
Tyson, J.J., Keener, J.P.: Singular perturbation theory of spiral waves in excitable media. Physica D 32, 327–361 (1989)
Keener, J.P.: The core of the spiral. SIAM J. Appl. Math. 52, 1370–1390 (1992)
Keener, J.P., Tyson, J.J.: The dynamics of scroll waves in excitable media. SIAM Rev. 34, 1–39 (1992)
Meron, E.: Pattern formation in excitable media. Phys. Rep. 218, 1–66 (1992)
Mikhailov, A.S.: Modeling pattern formation in excitable media: The Legacy of Norbert Wiener. In: Milton, J., Jung P. (eds.), Epilepsy as a Dynamic Disease. Berlin, Heidelberg, New York: Springer 2003
Mikhailov, A.S., Zykov, V.S.: Kinematic theory of spiral waves in excitable media: comparison with numerical simulations. Physica D 52, 379–397 (1991)
Plapp, B.B., Bodenschatz, E.: Core dynamics of multiarmed spirals in Rayleigh-Bénard convection. Phys. Scr. 67, 111–117 (1996)
Rudovics, B., de Kepper, P., Perraud, J.-J., Dulos, E.: Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system. Int. J. Bifurcat. Chaos 4, 1215–1231 (1994)
Rotermund, H.H., Nettesheim, S., von Oertzen, A., Ertl, G.: Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110) – front propagation and spiral waves. J. Chem. Phys. 98, 9977–9985 (1993)
Ungvarai-Nagy, Z., Ungvarai, J., Mueller, S.C.: Complexity in spiral wave dynamics. Chaos 3, 15–19 (1993)
Vanderbauwhede, A.: Centre manifolds, normal forms and elementary bifurcations. Dyn. Rep. 2, 89–169 (1989)
Winfree, A.T.: Spiral waves of chemical activity. Science 175, 634–636 (1972)
Winfree, A.T.: When Time Breaks down. Princeton, NJ: Princeton University Press 1987
Winfree, A.T.: The Geometry of Biological Time. Berlin, Heidelberg, New York: Springer 2001
Wiener, N., Rosenblueth, A.: 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, 205–265 (1946)
Zykov, V.S., Mikhailov, A.S., Mueller, S.C.: Wave instabilities in excitable media with fast inhibitor diffusion. Phys. Rev. Lett. 81, 2811–2814 (1998)
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Fiedler, B., Guo, JS. & Tsai, JC. Rotating spirals of curvature flows: a center manifold approach. Annali di Matematica 185 (Suppl 5), S259–S291 (2006). https://doi.org/10.1007/s10231-004-0145-1
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DOI: https://doi.org/10.1007/s10231-004-0145-1