Brusselator

{{Short description|Theoretical model for a type of autocatalytic reaction}}

File:Bruesselator.svg Bottom: The Brusselator in a stable regime with A=1 and B=1.7: For B<1+A² the system is stable and approaches a fixed point.]]

File:Brusselator space.gif

[[File:Brusselator Oscillations HQ Render.gif|thumb|Simulation{{cite web

| url = https://lukaswittmann.com/work/bachelors-thesis-the-brusselator/

| title = BACHELOR'S THESIS - DEVELOPMENT OF A PYTHON PROGRAM FOR INVESTIGATION OF REACTION-DIFFUSION COUPLING IN OSCILLATING REACTIONS

| author= Lukas Wittmann

}} of the reaction-diffusion system of the Brusselator with reflective border conditions]]

The Brusselator is a theoretical model for a type of autocatalytic reaction.

The Brusselator model was proposed by Ilya Prigogine and his collaborators at the Université Libre de Bruxelles.{{cite web

| url = http://www.idea.wsu.edu/OscilChem/#Brusselator%20Model

| title = IDEA - Internet Differential Equations Activities

| publisher = Washington State University

| accessdate = 2010-05-16

| archive-date = 2017-09-09

| archive-url = https://web.archive.org/web/20170909182522/http://www.idea.wsu.edu/OscilChem/#Brusselator%20Model

| url-status = dead

}}{{Cite journal |last=Prigogine |first=I. |last2=Lefever |first2=R. |date=1968-02-15 |title=Symmetry Breaking Instabilities in Dissipative Systems. II |url=https://pubs.aip.org/jcp/article/48/4/1695/83930/Symmetry-Breaking-Instabilities-in-Dissipative |journal=The Journal of Chemical Physics |language=en |volume=48 |issue=4 |pages=1695–1700 |doi=10.1063/1.1668896 |issn=0021-9606|url-access=subscription }}

It is a portmanteau of Brussels and oscillator.

It is characterized by the reactions

:$A\; \backslash rightarrow\; X$

:$2X\; +\; Y\; \backslash rightarrow\; 3X$

:$B\; +\; X\; \backslash rightarrow\; Y\; +\; D$

:$X\; \backslash rightarrow\; E$

Under conditions where A and B are in vast excess and can thus be modeled at constant concentration, the rate equations become

:$\{d\; \backslash over\; dt\}\backslash left\backslash \{\; X\; \backslash right\backslash \}\; =\; \backslash left\backslash \{A\; \backslash right\backslash \}\; +\; \backslash left\backslash \{\; X\; \backslash right\backslash \}^2\; \backslash left\backslash \{Y\; \backslash right\backslash \}\; -\; \backslash left\backslash \{B\; \backslash right\backslash \}\; \backslash left\backslash \{X\; \backslash right\backslash \}\; -\; \backslash left\backslash \{X\; \backslash right\backslash \}\; \backslash ,$

:$\{d\; \backslash over\; dt\}\backslash left\backslash \{\; Y\; \backslash right\backslash \}\; =\; \backslash left\backslash \{B\; \backslash right\backslash \}\; \backslash left\backslash \{X\; \backslash right\backslash \}\; -\; \backslash left\backslash \{\; X\; \backslash right\backslash \}^2\; \backslash left\backslash \{Y\; \backslash right\backslash \}\; \backslash ,$

where, for convenience, the rate constants have been set to 1.

The Brusselator has a fixed point at

:$\backslash left\backslash \{\; X\; \backslash right\backslash \}\; =\; A\; \backslash ,$

:$\backslash left\backslash \{\; Y\; \backslash right\backslash \}\; =\; \{B\; \backslash over\; A\}\; \backslash ,$.

The fixed point becomes unstable when

:$B>1+A^2\; \backslash ,$

leading to an oscillation of the system. Unlike the Lotka–Volterra equation, the oscillations of the Brusselator do not depend on the amount of reactant present initially. Instead, after sufficient time, the oscillations approach a limit cycle.http://www.bibliotecapleyades.net/archivos_pdf/brusselator.pdf Dynamics of the Brusselator

The best-known example is the clock reaction, the Belousov–Zhabotinsky reaction (BZ reaction). It can be created with a mixture of potassium bromate (KBrO₃), malonic acid (CH₂(COOH)₂), and manganese sulfate (MnSO₄) prepared in a heated solution of sulfuric acid (H₂SO₄).[http://online.redwoods.cc.ca.us/instruct/darnold/deproj/Sp98/Gabe/ BZ reaction] {{webarchive |url=https://web.archive.org/web/20121231011212/http://online.redwoods.cc.ca.us/instruct/darnold/deproj/Sp98/Gabe/ |date=December 31, 2012 }}