Instabilities of wave trains and turing patterns in large domains

International Journal of Bifurcation and Chaos

Volumn 17, Issue 8, 2007, Pages 2679-2691

  Rademacher, Jens D M ^a   Scheel, Arnd ^b  

^a CWI   (Netherlands)

^b UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Classification of instabilities; Reaction diffusion systems; Stability; Wave trains

Indexed keywords

MODULATION; PARAMETER ESTIMATION;

REACTION-DIFFUSION SYSTEMS; TURING PATTERNS; WAVE TRAINS;

SYSTEM STABILITY;

EID: 34748878233     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127407018683     Document Type: Article

References (41)

1. Bodenschatz, E., Pesch, W. & Ahlers, G. [2000] "Recent developments in Rayleigh-Bénard convection," Ann. Rev. Fluid Mech. 32, 709-778.
2. Briggs, R. J. [1964] Electron-Stream Interaction with Plasmas (MIT Press).
3. Castets, V., Dulos, E., Boissonade, J. & De Kepper, P. [1990] "Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern," Phys. Rev. Lett. 64, 2953-2956.
4. Chossat, P. & Iooss, G. [1994] The Couette-Taylor Problem (Springer, NY).
5. Ciocci, M.-C. & Vanderbauwhede, A. [2004] "Bifurcation of periodic points in reversible diffeomorphisms," in New Progress in Difference Equations, eds. Elaydi, A. S., Ladas, G. & Aulbach, B. (Chapman & Hall, CRC Press), pp. 75-93.
6. Cross, M. C. & Hohenberg, P. C. [1993] "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112.
7. De Wit, A. [1999] "Spatial patterns and spatiotemporal dynamics in chemical systems," Adv. Chem. Phys. 109, 435-513.
8. Doelman, A., Gardner, R. A. & Jones, C. K. R. T. [1995] "Instability of quasiperiodic solutions of the Ginzburg-Landau equation," Proc. Roy. Soc. Edinb. 125, 501-517.
9. Doelman, A., Sandstede, B., Scheel, A. & Schneider, G. [2006] "The dynamics of modulated wave trains," Memoirs of the AMS, to appear.
10. Fiedler, B. [1988] Global Bifurcation of Periodic Solutions with Symmetry, Lecture Notes in Mathematics, Vol. 1309 (Springer-Verlag, Heidelberg).
11. Fiedler, B. [1996] Global Pathfollowing of Homoclinic Orbits in Two-Parameter Flows, Pitman Res. Notes Math., Vol. 352, pp. 79-146.
12. Fiedler, B. & Heinze, S. [1996a] "Homotopy invariants of time reversible periodic orbits I: Theory," J. Diff. Eq. 126, 184-203.
13. Fiedler, B. & Heinze, S. [1996b] "Homotopy invariants of time reversible periodic orbits II: Applications," J. Diff. Eq. 126, 204-223.
14. Fiedler, B., Sandstede, B., Scheel, A. & Wulff, C. [1996] "Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders, and drifts," Documenta Mathematica 1, 479-505.
15. Fiedler, B. & Turaev, D. [1996] "Coalescence of reversible homoclinic orbits causes elliptic resonance, nonlinear dynamics, bifurcations and chaotic behavior," Int. J. Bifurcation and Chaos 6, 1007-1027.
16. Gardner, R. A. [1993] "On the structure of the spectra of periodic travelling waves," J. Math. Pures Appl. IX. 72, 415-439.
17. Golubitsky, M., Stewart, I. & Schaeffer, D. G. [1988] Singularities and Groups in Bifurcation Theory, Volume II, Applied Mathematical Sciences, Vol. 69 (Springer-Verlag, NY).
18. Golubitsky, M., LeBlanc, V. G., & Melbourne, I. [2000] "Hopf bifurcation from rotating waves and patterns in physical space," J. Nonlin. Sci. 10, 69-101.
19. Haragus, M., Lombardi, E. & Scheel, A. [2006] "Spectral stability of wave trains in the Kawahara equation," J. Math. Fluid Mech. 8, 482-509.
20. Haragus, M. & Scheel, A. [2006] "Corner defects in almost planar interface propagation," Ann. Inst. H. Poincare (C) Anal. Non Lineaire 23, 283-329.
21. Knobloch, J. & Vanderbauwhede, A. [1995] "Hopf bifurcation at k-fold resonances in reversible systems," Nonlinear Dynamical Systems and Chaos, eds. Broer et al. (Birkhäuser), pp. 155-171.
22. Krupa, M. [1990] "Bifurcations of relative equilibria," SIAM J. Math. Anal. 21, 1453-1486.
23. Mielke, A. [1997] "Instability and stability of rolls in the Swift-Hohenberg equation," Comm. Math. Phys. 189, 829-853
24. Mielke, A. [2002] "The Ginzburg-Landau equation in its role as a modulation equation," in Handbook of Dynamical Systems II (Elsevier Science B. V.), pp. 759-834.
25. Rademacher, J. D. M., Sandstede, B. & Scheel, A. [2007] "Computing absolute and essential spectra using continuation," Physica D 229, 166-183.
26. Rademacher, J. D. M. & Scheel, A. [2007] "The saddlenode of nearly homogeneous wave trains in reaction diffusion systems," J. Dyn. Diff. Eqns. 19, 479-496.
27. Sandstede, B. & Scheel, A. [2000a] "Absolute and convective instabilities of waves on unbounded and large bounded domains," Physica D 145, 233-277.
28. Sandstede, B. & Scheel, A. [2000b] "Spectral stability of modulated travelling waves bifurcating near essential instabilities," Proc. Roy. Soc. Edinb. A 130, 419-448.
29. Sandstede, B. & Scheel, A. [2001a] "On the structure of spectra of modulated travelling waves," Math. Nachr. 232, 39-93.
30. Sandstede, B. & Scheel, A. [2001b] "On the stability of periodic travelling waves with large spatial period," J. Diff. Eqns. 172, 134-188.
31. Sandstede, B. & Scheel, A. [2004] "Defects in oscillatory media: Toward a classification," SIAM J. Appl. Dyn. Syst. 3, 1-68.
32. Sandstede, B. & Scheel, A. [2007] "The period doubling of spiral waves," SIAM J. Appl. Dyn. Syst. 6, 494-547.
33. Scheel, A. Radially Symmetric Patterns of Reaction-Diffusion Systems, Mem. Amer. Math. Soc. Vol. 165.
34. Shil'nikov, L. P., Shil'nikov, A. L., Turaev, D. & Chua, L. O. [1998] Methods of Qualitative Theory in Nonlinear Dynamics, Part I (With the collaboration of Sergey Gonchenko (Sections 3.7 and 3.8), Oleg Stenckin (Section 3.9 and Appendix A) and Mikhail Shashkov (Sections 6.1 and 6.2) World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, Vol. 4.
35. Shil'nikov, L. P., Shil'nikov, A. L., Turaev, D. & Chua, L. O. [2001] Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, Vol. 5.
36. Turing, A. M. [1952] "The chemical basis of morphogenesis," Phil. Trans. R. Soc. London B 237, 37-72.
37. van Harten, A. [1995] "Modulated modulation equations," Proc. IUTAM/ISIMM Symp. Structure and Dynamics of Nonlinear Waves in Fluids (Hannover, 1994), Adv. Ser. Nonlinear Dynam., Vol. 7 (World Scientific, Singapore), pp. 117-130.
38. Vanderbauwhede, A. [1982] Local Bifurcation and Symmetry, Research Notes in Mathematics, Vol. 75 (Boston-London-Melbourne: Pitman Advanced Publishing Program. IV).
39. Vanderbauwhede, A. & Fiedler, B. [1992] "Homoclinic period blow-up in reversible and conservative systems," Z. Angew. Math. Phys. 43, 292-318.
40. Vanderbauwhede, A. [2001] "Subharmonic bifurcation at multiple resonances," Proc. Mathematics Conf. (Birzeit, August 1998), eds. Elaydi, S., Allen, F., Elkhader, A., Mughrabi, T. & Saleh, M. (World Scientific, Singapore), pp. 254-276.
41. Yoneyama, M., Fujii, A. & Maeda, S. [1995] "Wavelength-doubled spiral fragments in photosensitive monolayers," J. Am. Chem. Soc. 117, 8188-9191.