Hopf dances near the tips of busse balloons

Discrete and Continuous Dynamical Systems - Series S

Volumn 5, Issue 1, 2012, Pages 61-92

  Doelman, Arjen ^a,b   Rademacher, Jens D M ^b   Van Der Stelt, Sjors ^c  

^a LEIDEN UNIVERSITY   (Netherlands)

^b CWI   (Netherlands)

^c Korteweg de Vries Instituut   (Netherlands)

Author keywords

Numerical computation; Pulses; Reaction diffusion system; Semi strong limit; Stability boundaries; Wavetrains

Indexed keywords

EID: 84864488671     PISSN: 19371632     EISSN: 19371179     Source Type: Journal    
DOI: 10.3934/dcdss.2012.5.61     Document Type: Article

References (46)

1. I. Aranson and L. Kramer, The world of the Ginzburg-Landau equation, Rev. Modern Phys., 74 (2002), 99-143.
2. F. H. Busse, Nonlinear properties of thermal convection, Rep. Prog. Phys., 41 (1978), 1929-1967.
3. R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89-113.
4. E. J. Doedel. AUTO-07P: Continuation and bifurcation software for ordinary differential equations, http://cmvl.cs.concordia.ca/auto.
5. A. Doelman, An explicit theory for pulses in two component singularly perturbed reaction-diffusion equations, in preparation.
6. A. Doelman and W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations, Physica D, 53 (1991), 249-266.
7. A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach, Physica D, 122 (1998), 1-36. (Pubitemid 128636761)
8. A. Doelman, R.A. Gardner and T.J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Ind. Univ. Math. J., 50 (2001), 443-507. (Pubitemid 33589973)
9. A. Doelman, R. A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model, Memoirs AMS, 155 (2002),(737).
10. A. Doelman, T. J Kaper and H. van der Ploeg, Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer-Meinhardt equation, Meth. Appl. An., 8 (2001), 387-414. (Pubitemid 36299751)
11. A. Doelman, T.J. Kaper and P. Zegeling, Pattern formation in the one-dimensional Gray-Sc ott model, Nonlinearity, 10 (1997), 523-563. (Pubitemid 127430235)
12. A. Doelman and H. van der Ploeg, Homoclinic stripe patterns, SIAM J. Appl. Dyn. Syst., 1 (2002), 65-104. (Pubitemid 38818563)
13. A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains, Memoirs of the AMS, 199 (2009),(934).
14. W. Eckhaus and G. Iooss, Strong selection or rejection of spatially periodic patterns in degenerate bifurcations, Physica D, 39 (1989), 124-146.
15. E. G. Eszter, "Evans Function Analysis of the Stability of Periodic Traveling Wave Solutions of the Fitzhugh-Nagumo System," PhD thesis, U. of Massachusetts in Amherst, 1999.
16. R. A. Gardner, On the structure of the spectra of periodic travelling waves, J. Math. Pure Appl., 72 (1993), 415-439.
17. R. A. Gardner, Spectral analysis of long wavelength periodic waves and applications, J. Reine Angew. Math., 491 (1997), 149-181. (Pubitemid 127338563)
18. D. Iron and M. J. Ward, The dynamics of multi-spike solutions to the one-dimensional Giere r-Meinhardt model, SIAM J. Appl. Math., 62 (2002), 1924-1951.
19. D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62.
20. T. Kolokolnikov, M. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime, Physica D, 202 (2005), 258-293. (Pubitemid 40467090)
21. T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime, Stud. Appl. Math., 115 (2005), 21-71. (Pubitemid 40771805)
22. B. J. Matkowsky and V. A. Volpert, Stability of plane wave solutions of complex Ginzburg-Landau equations, Quart. Appl. Math., 51 (1993), 265-281.
23. A. Mielke, The Ginzburg-Landau equation in its role as modulation equation, in "Handbook of Dynamical Systems, II" (ed. B. Fiedler), Elsevier,(2002), pp. 759-835.
24. D.S. Morgan, A. Doelman and T.J. Kaper, Stationary periodic patterns in the 1D Gray-Scott model, Meth. Appl. Anal., 7 (2002), 105-150.
25. C. B. Muratov and V. V. Osipov, Traveling spike autosolitons in the Gray-Scott model, Physica D, 155 (2001), 112-131. (Pubitemid 32499021)
26. C. Muratov and V. V. Osipov, Stability of the static spike autosolitons in the Gray-Scott model, SIAM J. Appl. Math., 62 (2002), 1463-1487. (Pubitemid 35298521)
27. Y. Nishiura and D. Ueyama, A skeleton structure for self-replication dynamics, Physica D, 130 (1999), 73-104.
28. Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model, Physica D, 150 (2001), 137-162. (Pubitemid 32265308)
29. W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices AMS, 45 (1998), 9-18.
30. M. Oh and K. Zumbrun, Stability of periodic solutions of conservation laws with viscosity: Analysis of the Evans function, Arch. Rational Mech. Anal., 166 (2003), 99-166. (Pubitemid 36760983)
31. J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192.
32. H. van der Ploeg and A. Doelman, Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations, Indiana Univ. Math. J., 54 (2005), 1219-1301. (Pubitemid 41730645)
33. V. Petrov, S. K. Scott and K. Showalter, Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system, Phil. Trans. Roy. Soc. Lond., Series A, 347 (1994), 631-642.
34. J. D. M. Rademacher, B. Sandstede and A. Scheel, Computing absolute and essential spectra using continuation, Physica D, 229 (2007), 166-183. (Pubitemid 46755396)
35. J. D. M. Rademacher and A. Scheel, Instabilities of wave trains and Turing patterns in large domains, Int. J. Bif. Chaos, 17 (2007), 2679-2691. (Pubitemid 47479148)
36. J. D. M. Rademacher and A. Scheel, The saddle-node of nearly homogeneous wave trains in reaction-diffusion systems, J. Dyn. Diff. Eq., 19 (2007), 479-496.
37. W. N. Reynolds, J. E. Pearson and S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction diffusion systems, Phys. Rev. Lett., 72 (1994), 2797-2800. (Pubitemid 24975262)
38. B. Sandstede, Stability of travelling waves, in "Handbook ofDynamicalSystems, II" (ed.B. Fiedler), Elsevier,(2002), 983-1055.
39. B. Sandstede and A. Scheel, Absolute and convective instabilities of waves on unbounded and large dounded domains, Physica D, 145 (2000), 233-277.
40. B. Sandstede and A. Scheel, On the stability of periodic travelling waves with large spatial period, J. Diff. Eq., 172 (2001), 134-188. (Pubitemid 33377866)
41. A. Shepeleva, On the validity of the degenerate Ginzburg-Landau equation, Math. Methods Appl. Sci., 20 (1997), 1239-1256. (Pubitemid 127600117)
42. A. Shepeleva, Modulated modulations approach to the loss of stability of periodic solutions for the degenerate Ginzburg-Landau equation, Nonlinearity, 11 (1998), 409-429. (Pubitemid 128554135)
43. M. J. Smith and J. A. Sherratt, The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems, Physica D, 236 (2007), 90-103. (Pubitemid 350026085)
44. M. J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model, J. Nonl. Sc., 13 (2003), 209-264.
45. 2, Physica D, 176 (2003), 147-180.
46. J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl.(9), 83 (2004), 433-476. (Pubitemid 38393993)