The amplitude equation of Brusselator

  • Thread starter kail85
  • Start date
  • Tags
    Amplitude
In summary, in order to investigate the behavior of the Brusselor reaction system beyond the instability, multiple-scale analysis is used with the expansion \bftext{X}(t) = \bftext{X_0} + \rho\bftext{X_1} + \rho^2\bftext{X_2}. At higher orders of \rho, the method of multiple scales can be used to determine the amplitude coefficients, but the accuracy of the results depends on choosing a suitable value for \rho.
  • #1
kail85
1
0
Brusselor reaction can be represented as kinetic equations:


dX/dt = A-(B+1)X + X^2Y
dY/dt = BX - X^2Y


in order to investigate the behavior of the system beyond tje instability(B is used as bifurcation parameter, the critical value is B_H= 1+A^2), I use multiple-scale analysis:



\bftext{X}(t) = (X(t), Y(t))



\bftext{X}(t) = \bftext{X_0} + +\rho\bftext{X_1} + \rho^2\bftext{X_2}



\rho is a very very value.



B = B_H + \rho B_1 + \rho^2 B_2 + ...



d_t = d_{\tau_0} + \rho d_{\tau_1} + \rho^2 d_{\tau_2} +...

in which \tau_i scales as \tau_i = \rho^i t



the solution \bftext{X_1} is a linear combination of the right eigenvectors of the Jacobian martrix L:



L=[B-1 A^2 ; -B -A^2]



\bftext{X_1} = (1; (i-A)/A)Z(t) exp(iAt) + (1; -(i+A)/A)Z^*(t) exp(-iAt)



in order to determine the amplitude Z(t) and Z^*(t), the higher order contribution in \rho should be considered.



Now I know at the order \rho^0, one recovers the reference state (X_0, Y_0)



at the order \rho^1, the linear stability analysis is recovered



So, what will happen at the order \rho^2 and \rho^3, how to identify the amplitude coefficient?? Thank you
 
Physics news on Phys.org
  • #2
for your question and for sharing your approach to investigating the behavior of the Brusselor reaction. It seems like you have a good understanding of multiple-scale analysis and its application in this system.

At the order of \rho^2, we can use the method of multiple scales to determine the amplitude coefficient. This involves substituting the expansion \bftext{X}(t) = \bftext{X_0} + \rho\bftext{X_1} + \rho^2\bftext{X_2} into the kinetic equations and equating terms of the same order in \rho. This will result in a system of equations that can be solved for the amplitude coefficients.

Similarly, at the order of \rho^3, we can again use the method of multiple scales to determine the amplitude coefficients. This process can be repeated for higher orders of \rho to obtain a more accurate approximation of the behavior of the system.

It is important to note that the accuracy of this approach relies on the assumption that \rho is a very small value. Therefore, it is important to carefully choose a suitable value for \rho in order to ensure the validity of the results.

I hope this helps answer your question. Good luck with your investigation!
 
  • #3
for your question. The amplitude equation for the Brusselator reaction can be derived using multiple-scale analysis. This method allows us to investigate the behavior of the system beyond its instability point, which is determined by the bifurcation parameter B.

To derive the amplitude equation, we first introduce a small parameter ρ and expand the solution \bftext{X}(t) in powers of ρ. This allows us to consider higher order contributions in ρ and determine the amplitude of the solution. We also expand the bifurcation parameter B in powers of ρ, B = B_H + ρB_1 + ρ^2B_2 + ... where B_H is the critical value of B at the instability point.

Next, we introduce a new time variable τ which scales as τ_i = ρ^i t, and then expand the time derivative d_t in powers of ρ. This leads to a set of equations for the amplitudes \bftext{X_i} that are solved using linear algebra techniques. The solution \bftext{X_1} is a linear combination of the right eigenvectors of the Jacobian matrix L, which contains the coefficients of the kinetic equations.

To determine the amplitude Z(t) and its complex conjugate Z^*(t), we need to consider higher order contributions in ρ. At the order ρ^0, we recover the reference state (X_0, Y_0) of the system. At the order ρ^1, we recover the linear stability analysis. However, at the order ρ^2 and ρ^3, we need to include higher order terms to determine the amplitude coefficients.

In summary, the amplitude equation for the Brusselator reaction can be derived using multiple-scale analysis, which allows us to investigate the behavior of the system beyond its instability point. This method is useful for understanding the dynamics of complex systems and can provide insight into their behavior.
 

1. What is the Brusselator equation and what does it model?

The Brusselator is a mathematical model that describes the dynamics of chemical reactions. It specifically models the behavior of a chemical system that exhibits oscillations or periodic behavior, such as the Belousov-Zhabotinsky reaction.

2. How is the Brusselator equation used in scientific research?

The Brusselator equation is used in scientific research to study the behavior of chemical reactions and understand how they can produce oscillations. It is also used to model other complex systems, such as biological systems, economic systems, and ecological systems.

3. What is the amplitude equation of the Brusselator and what does it represent?

The amplitude equation of the Brusselator is a simplified version of the original equation that focuses on the amplitude or the maximum value of the oscillations. It represents the behavior of the system near a critical point, where the oscillations are small and can be described by a linear equation.

4. How is the amplitude equation of the Brusselator derived?

The amplitude equation of the Brusselator is derived using a technique called perturbation analysis. This involves assuming that the system is close to a critical point and expanding the original equation in terms of small deviations from this point. The resulting equation is then simplified to focus on the amplitude of the oscillations.

5. What are some real-world applications of the Brusselator equation?

The Brusselator equation has been used to model various phenomena in the natural world, such as the behavior of chemical reactions, the dynamics of animal populations, and the spread of diseases. It has also been applied in engineering, such as in the design of chemical reactors and in the study of nonlinear systems in control theory.

Similar threads

  • Advanced Physics Homework Help
2
Replies
36
Views
2K
Replies
1
Views
683
  • Advanced Physics Homework Help
Replies
1
Views
2K
Replies
2
Views
4K
  • High Energy, Nuclear, Particle Physics
Replies
9
Views
2K
  • Advanced Physics Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
453
  • Classical Physics
Replies
9
Views
2K
  • Classical Physics
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
747
Back
Top