Hopf dances near the tips of Busse balloons
Pages: 61 - 92,
Volume 5,
Issue 1,
February 2012
doi:10.3934/dcdss.2012.5.61 Abstract
References
Full text (563.5K)
Related Articles
Arjen Doelman - Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, Netherlands (email)
Jens D. M. Rademacher - Centrum Wiskunde en Informatica (CWI), Science Park 123, 1098 XG Amsterdam, Netherlands (email)
Sjors van der Stelt - Korteweg-de Vries Instituut, Science Park 904, 1098 XH Amsterdam, Netherlands (email)
| 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. |
|
| 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. |
|
| 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. |
|
| 11 |
A. Doelman, T. J. Kaper and P. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563. |
|
| 12 |
A. Doelman and H. van der Ploeg, Homoclinic stripe patterns, SIAM J. Appl. Dyn. Syst., 1 (2002), 65-104. |
|
| 13 |
A. Doelman, B. Sandstede, A. Scheel and G. Schneider, The dynamics of modulated wave trains, Memoirs of the AMS, 199 (2009). |
|
| 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. |
|
| 18 |
D. Iron and M. J. Ward, The dynamics of multi-spike solutions to the one-dimensional Gierer-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. |
|
| 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. |
|
| 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. |
|
| 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. |
|
| 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. |
|
| 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. |
|
| 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. |
|
| 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. |
|
| 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. |
|
| 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. |
|
| 38 |
B. Sandstede, Stability of travelling waves, in "Handbook of Dynamical Systems, 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. |
|
| 41 |
A. Shepeleva, On the validity of the degenerate Ginzburg-Landau equation, Math. Methods Appl. Sci., 20 (1997), 1239-1256. |
|
| 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. |
|
| 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. |
|
| 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 |
J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in $\RR^2$, 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. |
|
Go to top
|