Chapter 7.5, Problem 20P

There are certain chemical reactions in which the constituent concentrations oscillate periodically over time. The system

$x\text{'}=1-\left(b+1\right)x+\frac{x^{2}y}{4},y\text{'}=bx-\frac{x^{2}y}{4}$

Is a special case of a model, known as the Brusselator, of this kind of reaction. Assume that $b$ is a positive parameter, and consider solutions in the first quadrant of the $xy-$ plane.

(a) Show that the only critical point is $\left(1,4b\right)$.

(b) Find the eigenvalues of the approximate linear system at the critical point.

(c) Classify the critical point as to type and stability. How does the classification depend on $b$?

(d) As $b$ increases through a certain value $b_0$, the critical point changes from asymptotically stable to unstable. What is that value $b_0$?

(e) Plot trajectories in the phase plane for values of $b$ slightly less than and slightly greater than $b_0$. Observe the limit cycle when $b>b_0$; the Brusselator has a Hopf bifurcation point at $b_0$.

(f) Plot trajectories for several values of $b>b_0$ and observe how the limit cycle deforms as $b$ increases.