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x = a = x + x²y y=b-x²y. Here a, b 0 are dimensionless parameters and x, y> 0 are dimensionless chemical concentra- tions. To see videos of similar chemical oscillators, search for "Briggs-Rauscher" or "Belousov- Zhabotinsky," for example. (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) Open up pplane 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 pointing into the polygon along each of its borders, which means a solution can enter, but cannot leave the region. Do this in parts: 1. Show that y> 0 on the bottom, and that y<0 on the (horizontal) top. 2. Show that > 0 on the left side and i <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.

x = a = x + x²y y=b-x²y. Here a, b 0 are dimensionless parameters and x, y> 0 are dimensionless chemical concentra- tions. To see videos of similar chemical oscillators, search for "Briggs-Rauscher" or "Belousov- Zhabotinsky," for example. (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) Open up pplane 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 pointing into the polygon along each of its borders, which means a solution can enter, but cannot leave the region. Do this in parts: 1. Show that y> 0 on the bottom, and that y<0 on the (horizontal) top. 2. Show that > 0 on the left side and i <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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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MAA2.1 Oscillating chemical reactions and Hopf bifurcations The chemical reaction X k₁ k_1 A B Y 2X + Y k3 x = a − x + x²y y=b-x²y. → 3X produces the following set of differential equations after applying the law of mass action and nondimensionalization: Here a, b>0 are dimensionless parameters and x, y > 0 are dimensionless chemical concentra- tions. To see videos of similar chemical oscillators, search for "Briggs-Rauscher" or "Belousov- Zhabotinsky," for example. (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) Open up pplane 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 pointing into the polygon along each of its borders, which means a solution can enter, but cannot leave the region. Do this in parts: 1. Show that y> 0 on the bottom, and that y < 0 on the (horizontal) top. 2. Show that > 0 on the left side and <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.
Transcribed Image Text:MAA2.1 Oscillating chemical reactions and Hopf bifurcations The chemical reaction X k₁ k_1 A B Y 2X + Y k3 x = a − x + x²y y=b-x²y. → 3X produces the following set of differential equations after applying the law of mass action and nondimensionalization: Here a, b>0 are dimensionless parameters and x, y > 0 are dimensionless chemical concentra- tions. To see videos of similar chemical oscillators, search for "Briggs-Rauscher" or "Belousov- Zhabotinsky," for example. (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) Open up pplane 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 pointing into the polygon along each of its borders, which means a solution can enter, but cannot leave the region. Do this in parts: 1. Show that y> 0 on the bottom, and that y < 0 on the (horizontal) top. 2. Show that > 0 on the left side and <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.
y |(9,5) (a +6, 5) b₂ line with slope - 1 X (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.
Transcribed Image Text:y |(9,5) (a +6, 5) b₂ line with slope - 1 X (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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