Brusselsator

The Brusselsator is a simple model for describing chemical oscillators . The Brusselsator was developed by Ilya Prigogine and René Lefever at the Université Libre de Bruxelles in Belgium, hence the name.

description

Solution of the Brusselsator for different boundary conditions , together with phase space plots. At the top there are stable oscillations, while in the lower case the solutions tend towards a fixed point.

Four hypothetical reaction equations form a simple model that reflects all the phenomena of chemical oscillators (such as the Belousov-Zhabotinsky reaction ). A similar model system was derived in 1985 at the Humboldt University in Berlin by simplifying a real existing reaction system.

I.	A.	→	X
II	B + X	→	Y + C
III	2X + Y	→	3X	( autocatalytic )
IV	X	→	D.
Σ (I-IV once each)	A + B	→	C + D

The reaction rates are reflected by the constants  k ₁ to  k ₄ , the concentrations of A and B are kept constant, and the products  C and D are continuously removed.

The concentrations of X and Y are sensitive to small perturbations and quickly reach an oscillating state if the overall reaction is far from equilibrium . So you have a thermodynamically open system and you can set up two rate equations for the concentration of X and Y:

${\displaystyle {\begin{alignedat}{2}&{\frac {\mathrm {d} X}{\mathrm {d} t}}=k_{1}A-&&k_{2}BX+k_{3}X^{2}Y-k_{4}X\\&{\frac {\mathrm {d} Y}{\mathrm {d} t}}=&&k_{2}BX-k_{3}X^{2}Y\end{alignedat}}}$

These differential equations can be solved numerically . The figure opposite shows some solutions. Depending on the choice of the free parameters k ₁ A , k ₂ B , k ₃ and k ₄ , different behavior results: in the upper case one sees stable oscillations, while in the lower case with a different choice of parameters the concentrations  X (t) and  Y ( t) strive for a fixed point in phase space .

Analysis of stability

As shown above, depending on the parameterization, the Brusselsator has stable oscillations as a solution or strives towards a fixed point in phase space.

This fixed point results from

${\displaystyle {\frac {\mathrm {d} X}{\mathrm {d} t}}={\frac {\mathrm {d} Y}{\mathrm {d} t}}=0}$

to:

${\displaystyle {\begin{alignedat}{2}\Rightarrow &X={\frac {k_{1}}{k_{4}}}&&\cdot A\\&Y={\frac {k_{4}\,k_{2}}{k_{3}\,k_{1}}}&&\cdot {\frac {B}{A}}\end{alignedat}}}$

With the help of the linear stability analysis it can also be shown that this fixed point becomes unstable if:

${\displaystyle B>{\frac {k_{4}}{k_{2}}}+{\frac {k_{3}}{k_{2}}}\cdot \left({\frac {k_{1}}{k_{4}}}\cdot A\right)^{2}}$

In this case the trajectories tend towards a limit cycle in phase space and the system carries out the oscillations shown. ${\displaystyle (X(t),Y(t))}$

As a reaction-diffusion model

A time and space-dependent cellular automaton of the Brusselsator with two source points and periodic boundary conditions. There are circular waves and spiral waves .

The model can also be extended to a reaction-diffusion model and, if the correct parameters are selected, chemical waves are obtained as a solution, as shown on the right.

The differential equations are a diffusion share or expanded and become: ${\displaystyle D_{X}\cdot \nabla ^{2}X({\vec {x}},t)}$ ${\displaystyle D_{Y}\cdot \nabla ^{2}Y({\vec {x}},t)}$

${\displaystyle {\begin{alignedat}{3}&{\frac {\partial X({\vec {x}},t)}{\partial t}}=k_{1}A&&-k_{2}BX({\vec {x}},t)+k_{3}X^{2}({\vec {x}},t)Y({\vec {x}},t)-k_{4}X({\vec {x}},t)&&&+D_{X}\cdot \nabla ^{2}X({\vec {x}},t)\\&{\frac {\partial Y({\vec {x}},t)}{\partial t}}=&&+k_{2}BX({\vec {x}},t)-k_{3}X^{2}({\vec {x}},t)Y({\vec {x}},t)&&&+D_{Y}\cdot \nabla ^{2}Y({\vec {x}},t)\end{alignedat}}}$

Is in here

• ${\displaystyle {\vec {x}}}$ a point in space
• ${\displaystyle D}$the diffusion coefficient
• ${\displaystyle \nabla ^{2}}$the Laplace operator , i.e. the sum of the second spatial derivatives in Cartesian coordinates .

literature

• Dilip Kondepudi, Ilya Prigogine: Modern Thermodynamics. From Heat Engines to Dissipative Structures. John Wiley & Sons, Weinheim, New York 1998.

Individual evidence

1. ↑ RJ Field: An oscillating response. In: Chemistry in our time, 7th year 1973, No. 6, pp. 171–176, doi: 10.1002 / ciuz.19730070603 .
2. ↑ K. Bar-Eli: The minimal bromate oscillator simplified , J. Phys. Chem., 1985, doi: 10.1021 / j100259a030 .
3. ↑ ^{a b c} Dilip Kondepudi, Prigogine : Modern Thermodynamics. From Heat Engines to Dissipative Structures. John Wiley & Sons, Chichester et al. 1998, ISBN 0-471-97393-9 .

Web links

• Jan Krieger: Reaction Diffusion Systems (PDF; 2.5 MB), 2004.