Numerical bifurcation and stability analysis of solitary pulses in an excitable reaction-diffusion medium

Computer Methods in Applied Mechanics and Engineering

Volumn 170, Issue 3-4, 1999, Pages 253-275

  Krishnan, J ^a   Kevrekidis, Ioannis G ^a   Or Guil, Michael ^b   Zimmerman, Martin G ^b,c   Bär, Markus ^b  

^a Princeton University   (United States)

^b MAX PLANCK INSTITUT FÜR PHYSIK KOMPLEXER SYSTEME   (Germany)

^c UNIVERSITAT DE LES ILLES BALEARS   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; BOUNDARY CONDITIONS; INTEGRATION; MATHEMATICAL MODELS; NUMERICAL ANALYSIS; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS;

EXCITABLE REACTION-DIFFUSION MEDIUM; FLOQUET ANALYSIS; SOLITARY PULSES;

BIFURCATION (MATHEMATICS);

EID: 0032623541     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(98)00198-4     Document Type: Article

References (49)

1. M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 (1993) 851.
2. R. Kapral and K. Showalter, eds., Chemical Waves and Patterns (Kluwer, Dordrecht, The Netherlands, 1995).
3. A.S. Mikhailov, Foundations of Synergetics I (Springer-Verlag, Berlin, 1990).
4. H. Engel, F.-J. Niedernostheide, H.-G. Purwins and E. Schöll, eds., Self Organisation in Activator-Inhibitor Systems: Semiconductors, Gas-Discharge and Chemical Active Media (Wissenschaft und Technik Verlag, Berlin, 1996).
5. A.T. Winfree, Varieties of spiral wave behaviour: An experimentalist's approach to the theory of excitable media, Chaos 1 (1991) 303.
6. J.J. Tyson and J.P. Keener, Singular perturbation theory of travelling waves in excitable media, Physica D 32 (1988) 327.
7. J.-L. Martiel and A. Goldbeter, A model based on receptor desensitization for cyclic AMP signaling in Dictyostelium cells, Biophys. J. 52 (1987) 897.
8. E. Meron, Pattern formation in excitable media, Phys. Rep. 218 (1992) 1.
9. M. Bär N. Gottschalk, M. Eiswirth and G. Ertl, Spiral waves in a surface reaction: Model calculations, J. Chem. Phys. 100 (1994) 1202.
10. M. Bär, M. Hildebrand, M. Eiswirth, M. Falcke, H. Engel and M. Neufeld, Chemical turbulence and standing waves in a surface reaction model: The influence of global coupling and wave instabilities, Chaos 4 (1994) 499.
11. M.G. Zimmermann, S.O. Firle, M.A. Natiello, M. Hildebrand, M. Eiswirth, M. Bär, A.K. Bangia and I.G. Kevrekidis, Pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction-diffusion model, Physica D 110 (1997) 290.
12. M. Kness, L. Tuckerman and D. Barkley, Symmetry breaking bifurcations in one-dimensional excitable media, Phys. Rev A 46 (1992) 5054.
13. A.R. Champneys and Y.A. Kuznetsov, Numerical detection and continuation of codimension-2 homoclinic bifurcations, Int. J. Bif. Chaos 4 (1994) 785.
14. H. Kokubu, Homoclinic and heteroclinic bifurcations of vectortields, Japan J. Appl. Math. 5 (1988) 455.
15. B. Sandstede, C.K.R.T. Jones and J.C. Alexander, Existence and stability of N-pulses on optical fibers with phase-sensitive amplifiers. Physica D 106 (1997) 167.
16. M.G. Zimmermann and M.A. Natiello, Global homoclinic and heteroclinic bifurcations close to a twisted heteroclinic cycle, Int. J. Bif. Chaos 8(1998).
17. M.D. Graham, I.G. Kevrekidis, K. Asakura, J. Lauterbach, K. Krischer, H.-H. Rotermund and G. Ertl, Effects of Boundaries on Pattern Formation: Catalytic Oxidation of CO on Platinum, Science 264 (1994) 80.
18. J.W. Evans, Nerve axon equations: 1 - Linear approximations; 2 - Stability at rest: 3 - Stability of the nerve impulse, Indiana Univ. Math. Jour. 21 ( 1972) 877; 22 (1972) 75; 22 (1972) 577.
19. J.W. Evans, Nerve axon equations: 1 - Linear approximations; 2 - Stability at rest: 3 - Stability of the nerve impulse, Indiana Univ. Math. Jour. 21 ( 1972) 877; 22 (1972) 75; 22 (1972) 577.
20. J.W. Evans, Nerve axon equations: 1 - Linear approximations; 2 - Stability at rest: 3 - Stability of the nerve impulse, Indiana Univ. Math. Jour. 21 ( 1972) 877; 22 (1972) 75; 22 (1972) 577.
21. R.L. Pego and M.I. Weinstein, Eigenvalues and instabilities of solitary waves, Phil. Trans. R. Soc. Lond. A 340 (1992) 47.
22. H.-C. Chang, E.A. Demekhin and D.I. Kopelevich, Local stability theory of solitary pulses in an active medium, Physica D 97 (1996) 353.
23. H.-C. Chang, E.A. Demekhin and D.I. Kopelevich, Laminarizing effects of dispersion in an active-dissipative nonlinear medium, Physica D 63 (1993) 299.
24. K. Krischer, M. Eiswirth and G. Ertl, Oscillatory CO Oxidation on Pt(110): Modeling of temporal self-organization. J. Chem. Phys. 96 (1992) 9161.
25. M. Bär and M. Eiswirth, Turbulence due to spiral wave breakup in a continuous excitable medium, Phys. Rev. E 48 (1993) R1635.
26. M. Hildebrand, M. Bär and M. Eiswirth, Statistics of topological defects and spatiotemporal chaos in a reaction diffusion system, Phys. Rev. Lett. 75 (1995) 1503.
27. J.H. Merkin, V. Petrov, S.K. Scott and K. Showalter, Wave-induced chemical chaos, Phys. Rev. Lett. 76 (1996) 546.
28. A. Karma, Electrical Altemans and spiral wave breakup in cardiac tissue. Chaos 4 (1994) 461.
29. W. Bykov, The bifurcations of separatrix contours and chaos, Physica D 62 (1993) 290.
30. P. Glendinning and C. Sparrow, T-points: A codimension two heteroclinic bifurcation, J. Stat. Phys. 43 (1986) 479.
31. E. Yanagida, Branching of double pulse solutions from single pulse solutions in nerve axon equations, J. Diff. Eq. 66 (1987) 243.
32. M. Kisaka, H. Kokubu and H. Oka, Bifurcations to N-homoclinic orbits and N-periodic orbits in vector fields. J. Dyn. Diff. Eqs. 5 (1993) 305.
33. S.-N. Chow, B. Deng and D. Terman, The bifurcation of homoclinic orbits from two heteroclinic orbits - A topological approach, Appl. Anal. 42 (1992) 275
34. S.-N. Chow, B. Deng and D. Terman, The bifurcation of homoclinic and periodic orbits from two heteroclinic orbits, SIAM J. Math. Anal. 21 (1990) 179.
35. R.L. Pego, P. Smereka and M.I. Weinstein, Oscillatory instability of travelling waves for a KdV-Burgers equation, Physica D 67 (1993) 45-65.
36. H.S. Brown and I.G. Kevrekidis, Modulated travelling waves in the Kuramolo Sivashinsky equation, Fields Institute Comm. 5 (1996) 45.
37. N.J. Balmforth, G.R. Ireley and R. Worthing, Pulse dynamics in an unstable medium, SIAM J. Appl. Math. 57 (1997) 205.
38. N.J. Balmforth, Solitary waves and homoclinic orbits, Ann. Rev. Fluid Mech. 27 (1995) 335.
39. X.-B. Lin, On using Melnikov's method for solving Silnikov's problem, Trans. Roy. Soc. Edinbrg. 116A (1990) 195.
40. E. Yanagida and K. Maginu, Stability of double-pulse solutions in nerve axon equations, SIAM J. Appl. Math. 49 (1989) 1.
41. B. Sandstede, Stability of multiple pulse solutions, Trans. Am. Math. Soc. 350 (1997) 1.
42. B. Sandstede, Stability of N-fronts bifurcating from a twisted heteroclinic loop and an application to the Fitzhugh-Nagumo equation, SIAM J. Math. Anal. 29 (1997) 183.
43. J. Christoph, M. Eiswirth, N. Hartmann, R. Imbihl, I.G. Kevrekidis and M. Bär, Anomalous dispersion and pulse interaction in an excitable surface reaction, Phys. Rev. Lett., submitted.
44. N. Hartmann, Zur oszillatorischen Dynamik der an Platinmetallen katalysierten Oxidation von Kohlenmonoxid mit Sauerstoff und Stickstoffmonoxid, Ph.D. Thesis, Free University Berlin, 1997.
45. E.J. Doedel, X.J. Wang and T.F. Fairgrieve, AUTO 94: Software for continuation and bifurcation in ordinary differential equations, Applied Mathematics Report, California Institute of Technology, 1994.
46. E.J. Doedel, M.J. Friedman and B.I. Kunin, Successive continuation for locating connecting orbits, Preprint.
47. A. Champneys, Y. Kuznetsov and B. Sandstede, A numerical toolbox for homclinic bifurcation analysis, Int. J. Bif. Chaos 6 (1996) 867.
48. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, Berlin, 1988).
49. A.K. Bangia, Instabilities and pattern formation in heterogeneous catalysis and fluid flow, Ph.D. Thesis, Princeton University, 1996.