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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]].]]
[[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]]

Revision as of 20:13, 14 July 2013

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.
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.[1] It is a portmanteau of Brussels and oscillator.

It is characterized by the reactions

with the rate equations

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

The Brusselator has a fixed point at

.

The fixed point becomes unstable when

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.[2]

The best-known example is the clock reaction, the Belousov–Zhabotinsky reaction (BZ reaction). It can be created with a mixture of potassium bromate , malonic acid , and manganese sulfate prepared in a heated solution of sulfuric acid .[3]

See also

References

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