×

Continuation of spiral waves. (English) Zbl 1114.35099

Summary: We describe a new numerical method of computing rigidly rotating spiral waves, which is based on solving the Neumann boundary-value problem for the radius-dependent angular Fourier modes. Utilizing the established continuation engine AUTO, our method is simple in implementation and can be easily modified to suit a particular reaction-diffusion system. Since the method does not involve direct simulations of the reaction-diffusion system, unstable branches of rigidly rotating spiral waves can be computed as well. We illustrate our method by computing single- and multi-armed spirals in the Barkley model. Continuation of single-armed spirals displays nearly identical results with the Barkley’s continuation code STEADY. The dependence of spiral waves on the geometry of the medium reproduces the results of numerical simulations reported before, revealing, however, some subtle details like non-monotonous dependence of the rotation frequency on the disc radius and the existence of an unstable rotating solution that separates coexisting free and pinned spirals. We demonstrate that on bounded discs, spiral waves are accompanied by boundary spots - slowly rotating solutions which are localized near the outer boundary of the disc. Boundary spots are shown to be closely related to one- and two-dimensional unstable critical solutions, such as unstable pulses in one dimension and critical fingers in two dimensions, which separate spiral waves from shrinking wave segments.

MSC:

35K57 Reaction-diffusion equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pertsov, A. M.; Ermakova, E. A.; Panfilov, A. V., Rotating spiral waves in a modified Fitz-Hugh-Nagumo model, Physica D, 14, 1, 117-124 (1984) · Zbl 0587.35080
[2] Cross, M.; Hohenberg, P. C., Pattern formation outside of equilibrium, Rev. Modern Phys., 65, 851-1112 (1993) · Zbl 1371.37001
[3] (Kapral, R.; Showalter, K., Chemical Waves and Patterns (1995), Kluwer Academic Publishers)
[4] Wilkins, M.; Sneyd, J., Intercellular spiral waves of calcium, J. Theoret. Biol., 191, 299-398 (1998)
[5] Barkley, D., A model for fast computer simulation of waves in excitable media, Physica D, 49, 61-70 (1991)
[6] ten Tusscher, K.; Noble, D.; Noble, P.; Panfilov, A., A model for human ventricular tissue, Amer. J. Physiol., 286, H1573-H1589 (2004)
[7] Wheeler, P.; Barkley, D., Computation of spiral spectra, SIAM J. Appl. Dyn. Syst., 5, 157-177 (2006) · Zbl 1093.35014
[8] Schlesner, J.; Zykov, V.; Engel, H.; Schöll, E., Stabilization of unstable rigid rotation of spiral waves in excitable media, Phys. Rev. E, 74, 4, 046215 (2006)
[9] Doedel, E.; Paffenroth, R.; Champneys, A.; Fairgrieve, T.; Kuznetsov, Y.; Oldeman, B.; Sandstede, B.; Wang, X., AUTO2000: Continuation and bifurcation software for ordinary differential equations (with HOMCONT) (2002), Concordia University: Concordia University Montreal
[10] D. Barkley, STEADY: Code for computing steady spirals and straight-twisted scroll waves, unpublished; D. Barkley, STEADY: Code for computing steady spirals and straight-twisted scroll waves, unpublished
[11] Barkley, D., Linear stability analysis of rotating spiral waves in excitable media, Phys. Rev. Lett., 68, 13, 2090-2093 (1992)
[12] Barkley, D., Spiral meandering, (Kapral, R.; Showalter, K., Chemical Waves and Patterns (1995), Kluwer), 163
[13] Bär, M.; Bangia, A. K.; Kevrekidis, I. G., Bifurcation and stability analysis of rotating chemical spirals in circular domains: Boundary-induced meandering and stabilization, Phys. Rev. E, 67, 5, 056126 (2003)
[14] Henry, H.; Hakim, V., Linear stability of scroll waves, Phys. Rev. Lett., 85, 25, 5328-5331 (2000)
[15] Echebarria, B.; Hakim, V.; Henry, H., Nonequilibrium ribbon model of twisted scroll waves, Phys. Rev. Lett., 96, 9, 098301 (2006)
[16] Laing, C. R., Spiral waves in nonlocal equations, SIAM J. Appl. Dyn. Syst., 4, 3, 588-606 (2005) · Zbl 1090.37056
[17] D. Lloyd, A. Champneys, Efficient numerical continuation and stability analysis of spatiotemporal quadratic optical solitons, SIAM J. Sci. Comput. (submitted for publication). http://www.enm.bris.ac.uk/anm/preprints/2004r12.pdf; D. Lloyd, A. Champneys, Efficient numerical continuation and stability analysis of spatiotemporal quadratic optical solitons, SIAM J. Sci. Comput. (submitted for publication). http://www.enm.bris.ac.uk/anm/preprints/2004r12.pdf · Zbl 1096.78006
[18] J. Rademacher, B. Sandstede, A. Scheel, Computing absolute and essential spectra using continuation. http://www.wias-berlin.de/people/rademach/specnum-subm.pdf; J. Rademacher, B. Sandstede, A. Scheel, Computing absolute and essential spectra using continuation. http://www.wias-berlin.de/people/rademach/specnum-subm.pdf · Zbl 1119.65114
[19] Aranson, I. S.; Kramer, L., The world of the complex Ginzburg-Landau equation, Rev. Modern. Phys., 74, 1, 99 (2002) · Zbl 1205.35299
[20] M. Tsoi, Ph.D. Thesis, University of Ohio http://www.ohiolink.edu/etd/view.cgi?osu1148486634; M. Tsoi, Ph.D. Thesis, University of Ohio http://www.ohiolink.edu/etd/view.cgi?osu1148486634
[21] Frigo, M.; Johnson, S. G., The design and implementation of FFTW3 (Program generation, optimization, and platform adaptation), Proc. IEEE, 93, 2, 216-231 (2005), (special issue)
[22] Hartmann, N.; Bär, M.; Kevrekidis, I. G.; Krischer, K.; Imbihl, R., Rotating chemical waves in small circular domains, Phys. Rev. Lett., 76, 8, 1384-1387 (1996)
[23] Davydov, V.; Zykov, V., Spiral autowaves in a round excitable medium, Zh. Eksp. Teor. Fiz, 103, 844-856 (1993)
[24] Manz, N.; Müller, S. C.; Steinbock, O., Anomalous dispersion of chemical waves in a homogeneously catalyzed reaction system, J. Phys. Chem. A, 104, 5895 (2000)
[25] Manz, N.; Hamik, C. T.; Steinbock, O., Tracking waves and vortex nucleation in excitable systems with anomalous dispersion, Phys. Rev. Lett., 92, 248301 (2004)
[26] Vanag, V. K.; Epstein, I. R., Packet waves in a reaction-diffusion system, Phys. Rev. Lett., 88, 088303 (2002)
[27] Christoph, J.; Eiswirth, M.; Hartmann, N.; Imbihl, R.; Kevrekidis, I.; Bär, M., Anomalous dispersion and pulse interaction in an excitable surface reaction, Phys. Rev. Lett., 82, 7, 1586-1589 (1999)
[28] Or-Guil, M.; Kevrekidis, I. G.; Bär, M., Stable bound states of pulses in an excitable medium, Physica D, 135, 154-174 (2000) · Zbl 0945.34032
[29] V. Zykov, Selection mechanism for rotating patterns in weakly excitable medium. Preprint; V. Zykov, Selection mechanism for rotating patterns in weakly excitable medium. Preprint
[30] Karma, A., Universal limit of spiral wave propagation in excitable media, Phys. Rev. Lett., 66, 17, 2274-2277 (1991) · Zbl 0968.35503
[31] Mihaliuk, E.; Sakurai, T.; Chirila, F.; Showalter, K., Feedback stabilization of unstable propagating waves, Phys. Rev. E, 65, 6, 065602 (2002)
[32] Zykov, V. S.; Showalter, K., Wave front interaction model of stabilized propagating wave segments, Phys. Rev. Lett., 94, 6, 068302 (2005)
[33] Mikhailov, A. S., Foundation of Synergetics I: Distributed Active Systems (1991), Springer: Springer Berlin
[34] Krupa, M.; Sandstede, B.; Szmolyan, P., Fast and slow waves in the FitzHugh-Nagumo equation, J. Differential Equations, 133, 49-97 (1997) · Zbl 0898.34050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.