Numerical Studies of Symmetry-Breaking Bifurcations in Reaction-Diffusion Systems

1988 ◽  
pp. 439-448 ◽  
Author(s):  
K. Duncan ◽  
J. C. Eilbeck
Keyword(s):  
Symmetry Breaking ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Numerical Studies ◽  
Diffusion Systems

scholarly journals Recent advances in the temporal and spatiotemporal dynamics induced by bromate–sulfite-based pH-oscillators

2021 ◽  
Author(s):  
István Szalai ◽  
Brigitta Dúzs ◽  
István Molnár ◽  
Krisztina Kurin-Csörgei ◽  
Miklós Orbán
Keyword(s):  
Reaction Diffusion ◽  
Spatiotemporal Dynamics ◽  
Physical Processes ◽  
Oxidation Number ◽  
Adsorption Ability ◽  
Chemical Oscillators ◽  
Reaction Diffusion Systems ◽  
Numerical Studies ◽  
Diffusion Systems ◽  
Diffusion Phenomena

AbstractThe bromate–sulfite reaction-based pH-oscillators represent one of the most useful subgroup among the chemical oscillators. They provide strong H+-pulses which can generate temporal oscillations in other systems coupled to them and they show wide variety of spatiotemporal dynamics when they are carried out in different gel reactors. Some examples are discussed. When pH-dependent chemical and physical processes are linked to a bromate–sulfite-based oscillator, rhythmic changes can appear in the concentration of some cations and anions, in the distribution of the species in a pH-sensitive stepwise complex formation, in the oxidation number of the central cation in a chelate complex, in the volume or the desorption-adsorption ability of a piece of gel. These reactions are quite suitable for generating spatiotemporal patterns in open reactors. Many reaction–diffusion phenomena, moving and stationary patterns, have been recently observed experimentally using different reactor configurations, which allow exploring the effect of different initial and boundary conditions. Here, we summarize the most relevant aspects of these experimental and numerical studies on bromate–sulfite reaction-based reaction–diffusion systems.

scholarly journals From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ

2020 ◽  
Vol 17 (162) ◽  
pp. 20190621 ◽  
Author(s):  
Andrew L. Krause ◽  
Václav Klika ◽  
Thomas E. Woolley ◽  
Eamonn A. Gaffney
Keyword(s):  
Symmetry Breaking ◽  
Heterogeneous Media ◽  
Heterogeneous Reaction ◽  
Reaction Diffusion ◽  
Spatial Patterning ◽  
Instability Mechanism ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Turing Instabilities ◽  
Pattern Forming

Pattern formation from homogeneity is well studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is non-trivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction–diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions. We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction–diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing’s original thesis to a far wider and more realistic class of systems.

EXAMPLES OF FORCED SYMMETRY-BREAKING TO HETEROCLINIC CYCLES AND NETWORKS IN THREE-DIMENSIONAL EUCLIDEAN-INVARIANT SYSTEMS

2009 ◽  
Vol 19 (05) ◽  
pp. 1655-1678
Author(s):  
M. J. PARKER ◽  
M. G. M. GOMES ◽  
I. N. STEWART
Keyword(s):  
Symmetry Breaking ◽  
Nonlinear Optical ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Three Dimensions ◽  
Optical Systems ◽  
Heteroclinic Cycles ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Face Centered

In [Parker et al., 2008a] group theory was employed to prove the existence of homoclinic cycles in forced symmetry-breaking of simple (SC), face-centered (FCC), and body-centered (BCC) cubic planforms. In this paper we extend this classification demonstrating that more elaborate heteroclinic cycles and networks can arise through the same process. Our methods naturally generate graphs that represent possible heteroclinic cycles and networks. The results do not depend on the representation of the symmetry group and are thus quite general. This study is motivated by pattern formation in three dimensions which occur in reaction–diffusion systems, certain nonlinear optical systems and the polyacrylamide methylene blue oxygen reaction. This work extends previous work by Parker et al. [2006, 2008a, 2008b] and Hou and Golubitsky [1997].

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