x' = a - x + (x^2)y y' = b - (x^2)y Here a, b > 0 are dimensionless parameters and x, y > 0 are dimensionless chemical concentra- tions. (a) This system has a single fixed point. Find it. To analyze stability, find the condition on a, b which separates the stability types. (b) Plot phase portrait and convince yourself that a stable limit cycle appears as soon as the fixed point become unstable. This is a Hopf bifurcation. (c) We'll now verify the above observation by showing that the polygon pictured on the back of this sheet is a trapping region. To do this, you should show that (x', y') is
x' = a - x + (x^2)y
y' = b - (x^2)y
Here a, b > 0 are dimensionless parameters and x, y > 0 are dimensionless chemical concentra-
tions.
(a) This system has a single fixed point. Find it. To analyze stability, find the condition on
a, b which separates the stability types.
(b) Plot phase portrait and convince yourself that a stable limit cycle appears as soon as the fixed
point become unstable. This is a Hopf bifurcation.
(c) We'll now verify the above observation by showing that the
of this sheet is a trapping region. To do this, you should show that (x', y') is pointing into
the polygon along each of its borders, which means a solution can enter, but cannot leave
the region. See attached image.
Do this in parts:
1. Show that y' > 0 on the bottom, and that y' < 0 on the (horizontal) top.
2. Show that x' > 0 on the left side and x' < 0 on the (vertical) right side.
3. Show that (x', y') * (1, 1) < 0 along the diagonal side.
4. Explain why, if the fixed point is unstable, there must be a stable limit cycle.
(d) Sketch a stability diagram: plot the region in the a, b plane in which there is a stable limit
cycle (and an unstable fixed point) and the region in which there is just a stable fixed point.
The border between these regions is a curve in a, b space on which Hopf bifurcations occur.
Use Mathematica for help plotting this curve.
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