Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T22:19:07.010Z Has data issue: false hasContentIssue false
  • English
  • Français

Moving and Breathing Localized Structures in Reaction-diffusion Systems

Published online by Cambridge University Press:  17 September 2013

S.V. Gurevich  and
R. Friedrich
S.V. Gurevich*
Affiliation:
Institut for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, 48149, Münster, Germany
R. Friedrich
Affiliation:
Institut for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, 48149, Münster, Germany
*
⋆ Corresponding author. E-mail: gurevics@uni-muenster.de
Get access

Abstract

We are interested in the stability of the localized stationary solutions of a three-component reaction-diffusion system with one activator and two inhibitors. We show that depending on control parameters, solutions in form of moving and breathing localized structures can be observed in the vicinity of the codimension-two bifurcation point. We analyze this situation performing multiple scale perturbation expansion in the vicinity of the bifurcation point and derive a set of order parameter equations, explicitly describing the dynamics of the single localized structure. Numerical simulations are carried out, showing good agreement with the analytical predictions.

Keywords

complex systemsdissipative solitonsbifurcation analysis
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 84 - 94
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Woesler, R., Schütz, P., Bode, M., Or-Guil, M., Purwins, H.-G.. Oscillations of fronts and front pairs in two- and three-component reaction-diffusion systems. Physica D, 91 (1996), 376–405, 1996. CrossRefGoogle Scholar
Schenk, C. P., Or-Guil, M., Bode, M., Purwins, H.-G.. Interacting pulses in three-component reaction diffusion systems on two-dimensional domains. Physical Review Letters, 78 (1997), 3781–3784. CrossRefGoogle Scholar
R. Kapral, K. Showalter, editors. Chemical Waves and Patterns, vol. 10 of Understanding Chemical Reactivity, Kluwer Academic Publishers, Dordrecht, 1995.
Dahlem, M. A., Graf, R., Strong, A. J., Dreier, J. P., Dahlem, Y. A., Sieber, M., Hanke, W., Podoll, K., Schöll, E.. Two-dimensional wave patterns of spreading depolarization: Retracting, re-entrant, and stationary waves. Physica D: Nonlinear Phenomena, 239 (2010), 889–903. CrossRefGoogle Scholar
Purwins, H.-G., Bïdeker, H.U.. Amiranashvili, Sh.. Dissipative solitons. Advances in Physics, 59 (2010), 485–701. CrossRefGoogle Scholar
H. Engel, F.-J. Niedernostheide, H.-G. Purwins, E. Schöll. Self-Organization in Activator-Inhibitor-Systems: Semiconductors, Gas-Discharge and Chemical Active Media. Wissenschaft und Technik Verlag, Berlin, 1996.
Gurevich, S. V., Amiranashvili, Sh., Purwins, H.-G.. Breathing dissipative solitons in three-component reaction-diffusion system. Physical Review E, 74 (2006) 066201. CrossRefGoogle ScholarPubMed
Or-Guil, M., Bode, M., Schenk, C. P., Purwins, H.-G.. Spot Bifurcations in Three-Component Reaction-Diffusion Systems: The Onset of Propagation. Physical Review E, 57 (1998), 6432–6437. CrossRefGoogle Scholar
Bode, M., Liehr, A. W., Schenk, C. P., Purwins, H.-G.. Interaction of Dissipative Solitons: Particle-like Behaviour of Localized Structures in a Three-Component Reaction-Diffusion System. Physica D, 161 (2002), 45–66. CrossRefGoogle Scholar
Bode, M.. Front-bifurcations in reaction-diffusion systems with inhomogeneous parameter distribution. Physica D, 106 (1997), 270–286. CrossRefGoogle Scholar
Schütz, P., Bode, M., Gafiichuk, V. V.. Transition from stationary to travelling localized patterns in a two-dimensional reaction-diffusion systems. Physical Review E, 52 (1995), 4465–4473 CrossRefGoogle Scholar
A. Ankiewicz N. Akhmediev, editors. Dissipative Solitons. Volume 661 of Lecture Notes in Physics, Springer, Berlin, 2005.
A. Ankiewicz N. Akhmediev, editors. Dissipative Solitons: From Optics to Biology and Medicine. Volume 751 of Lecture Notes in Physics, Springer, Berlin, 2008.
Mikhailov, A. S., Showalter, K.. Control of waves, patterns and turbulence in chemical systems. Physics Reports, 425 (2006), 79–194. CrossRefGoogle Scholar
Christov, C.I., Velarde, M.G.. Dissipative solitons. Physica D: Nonlinear Phenomena, 86 (1995), 323–347. CrossRefGoogle Scholar
R. Richter, A. Lange. Surface Instabilities of Ferrofluids. in S. Odenbach (editor), Colloidal Magnetic Fluids, vol. 763 of of Lecture Notes in Physics, Springer Berlin, 2009.
Lioubashevski, O., Hamiel, Y., Agnon, A., Reches, Z., Fineberg, J.. Oscillons and propagating solitary waves in a vertically vibrated colloidal suspension. Physical Review Letters, 83 (1999), 3190. CrossRefGoogle Scholar
Umbanhowar, P. B., Melo, F., Swinney, H. L.. Localized excitations in a vertically vibrated granular layer. Nature, 382 (1996), 793–796 CrossRefGoogle Scholar
H. Haken. Synergetics. Introduction and Advanced Topics, Springer, Berlin, 1983.