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[[File:Bruesselator.svg|thumb|right|350px|Top: The Brusselator in the unstable regime (A=1, B=3): The system approaches a [[limit cycle]] Bottom: The Brüsselator in a stable regime with A=1 and B=1.7: For B<1+A the system is stable and approaches a [[Fixed point (mathematics)|fixed point]].]]
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[[File:Brusselator space.gif|thumb|Simulation of the brusselator as reaction diffusion system in two spacial dimensions]]
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 [[Free University of Brussels]].<ref>{{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
}}</ref>
It is a portmanteau of [[Brussels]] and [[oscillator]].
 
It is characterized by the reactions
 
:<math>  A \rightarrow X</math>
 
:<math>  2X + Y \rightarrow 3X</math>
 
:<math>  B + X \rightarrow Y + D</math>
 
:<math>  X \rightarrow E</math>
 
with the rate equations
 
:<math>{d \over dt}\left\{ X \right\} = \left\{A \right\} + \left\{ X \right\}^2 \left\{Y \right\}  - \left\{B \right\} \left\{X \right\} - \left\{X \right\}  \,</math>
 
:<math>{d \over dt}\left\{ Y \right\} =  \left\{B \right\} \left\{X \right\} - \left\{ X \right\}^2 \left\{Y \right\}    \,</math>
 
where, for convenience, the rate constants have been set to 1.
 
The Brusselator has a fixed point at
 
:<math>\left\{ X \right\} =  A    \,</math>
 
:<math>\left\{ Y \right\} =  {B \over A}    \,</math>.
 
The fixed point becomes unstable when
 
:<math> B>1+A^2  \,</math>
 
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]].<ref>http://www.bibliotecapleyades.net/archivos_pdf/brusselator.pdf Dynamics of the Brusselator
</ref>
 
The best-known example is the [[clock reaction]], the [[Belousov–Zhabotinsky reaction]] (BZ reaction). It can be created with a mixture of potassium bromate <math>(KBrO_3)</math>, malonic acid <math>(CH_2(COOH)_2)</math>, and manganese sulfate <math>(MnSO_4)</math> prepared in a heated solution of sulfuric acid <math>(H_2SO_4)</math>.<ref>[http://online.redwoods.cc.ca.us/instruct/darnold/deproj/Sp98/Gabe/ BZ reaction]</ref>
 
==See also==
*[[Lotka–Volterra equation]]
*[[Oregonator]]
 
==References==
{{reflist}}
 
[[Category:Non-equilibrium thermodynamics]]

Latest revision as of 07:16, 8 December 2014

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