There are certain chemical reactions in which the consti
Question and Solution
There are certain chemical reactions in which the constituent concentrations oscillate periodically over time. The syste
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There are certain chemical reactions in which the constituent concentrations oscillate periodically over time. The system x? = 1 ? (b + 1)x + x2y 4 , y? = bx ? x2y 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 (1, 4b). (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 b0, the critical point changes from asymptotically stable to unstable. What is that value b0? (e) Plot trajectories in the phase plane for values of b slightly less than and slightly greater than b0. Observe the limit cycle when b > b0; the Brusselator has a Hopf bifurcation point at b0. (f) Plot trajectories for several values of b > b0 and observe how the limit cycle deforms as b increases.
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