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Scroll waves in isotropic excitable media: linear instabilities, bifurcations, and restabilized states. (English) Zbl 1244.35071
Summary: Scroll waves are three-dimensional analogs of spiral waves. The linear stability spectrum of untwisted and twisted scroll waves is computed for a two-variable reaction-diffusion model of an excitable medium. Different bands of modes are seen to be unstable in different regions of parameter space. The corresponding bifurcations and bifurcated states are characterized by performing direct numerical simulations. In addition, computations of the adjoint linear stability operator eigenmodes are also performed and serve to obtain a number of matrix elements characterizing the long-wavelength deformations of scroll waves.

MSC:
35K57 Reaction-diffusion equations
35B25 Singular perturbations in context of PDEs
35B32 Bifurcations in context of PDEs
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[1] V.S. Zykov, Biofizika 31 pp 862– (1986)
[2] A.T. Winfree, Chaos 1 pp 303– (1991) · Zbl 1031.76502
[3] A. Belmonte, J. Phys. II 7 pp 1425– (1997)
[4] A.T. Winfree, Science 266 pp 1003– (1994)
[5] A.M. Pertsov, Nature (London) 345 pp 419– (1990)
[6] S. Mironov, J. Phys. Chem. 100 pp 1975– (1996)
[7] A.T. Winfree, Chaos 6 pp 617– (1996)
[8] C. Henze, Can. J. Phys. 68 pp 683– (1989) · Zbl 0991.92501
[9] A.V. Panfilov, Physica D 28 pp 215– (1987)
[10] V.N. Biktashev, Philos. Trans. R. Soc. London, Ser. A 347 pp 611– (1994) · Zbl 0862.92002
[11] Z. Qu, Phys. Rev. Lett. 83 pp 2668– (1999)
[12] V. Hakim, Phys. Rev. E 60 pp 5073– (1999)
[13] I. Aranson, Phys. Rev. E E58 pp 4556– (1998)
[14] J.P. Keener, Physica D 31 pp 269– (1988) · Zbl 0645.76052
[15] H. Henry, Phys. Rev. Lett. 85 pp 5328– (2000)
[16] R. FitzHugh, Biophys. J. 1 pp 445– (1961)
[17] J. Nagumo, Proc. IRE 50 pp 2061– (1962)
[18] D. Barkley, Physica D 49 pp 61– (1991)
[19] I. Goldhirsch, J. Sci. Comput. 2 pp 33– (1987) · Zbl 0666.65033
[20] D. Barkley, Phys. Rev. Lett. 68 pp 2090– (1992)
[21] D. Barkley, Phys. Rev. Lett. 72 pp 164– (1994)
[22] A. Karma, Proc. Natl. Acad. Sci. U.S.A. 97 pp 5687– (2000)
[23] I. Mitkov, Phys. Rev. E 52 pp 5974– (1995)
[24] V. Krinsky, Phys. Rev. Lett. 76 pp 3854– (1996)
[25] I.S. Aranson, Phys. Rev. E 57 pp 5276– (1998)
[26] M. Gabbay, Physica D 118 pp 371– (1998) · Zbl 1194.35437
[27] G. Rousseau, Phys. Rev. Lett. 80 pp 5671– (1998)
[28] D. Margerit, Phys. Rev. Lett. 86 pp 175– (2001)
[29] A.V. Panfilov, Physica D 84 pp 545– (1995)
[30] F. Fenton, Phys. Rev. Lett. 81 pp 481– (1998)
[31] F. Fenton, Chaos 8 pp 20– (1998) · Zbl 1069.92503
[32] W.-J. Rappel, Chaos 11 pp 71– (2001) · Zbl 1070.92510
[33] S. Setayeshgar, Phys. Rev. Lett. 88 pp 028101– (2002)
[34] W.T. Baxter, Biophys. J. 80 pp 516– (2001)
[35] M. Dowle, Int. J. Bifurcation Chaos Appl. Sci. Eng. 7 pp 2529– (1997) · Zbl 0899.92002
[36] K.I. Agladze, J. Phys. Chem. 96 pp 5239– (1992)
[37] O. Steinbock, Phys. Rev. Lett. 68 pp 248– (1992)
[38] A. Belmonte, Europhys. Lett. 32 pp 267– (1995)
[39] J.H. White, Am. J. Math. 91 pp 693– (1969) · Zbl 0193.50903
[40] W.F. Pohl, Math. Intell. 3 pp 20– (1980) · Zbl 0447.92013
[41] F.B. Fuller, Proc. Natl. Acad. Sci. U.S.A. 75 pp 3557– (1978) · Zbl 0395.92010
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