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hw6 - Hint for 4.3.7 For this problem the bifurcations...

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Math 454 Problem Set #6 Fall 2008 Sect. 1 Due Tuesday, 10/7, at the beginning of class. Reminder. You are encouraged to work together on homework problems. However, the final write- up must be your own. You will be quizzed on some of these problems, so be sure you can do all the problems without help. Read: Sects. 4.5, 5.1-2 Problems - Sect. 4.3 #2, 7*, 8* (bonus) - Sect. 4.5 #1(a-c) * The vector fields in 4.3.7 & 8 can have “singularities,” that is, places where the vector field is infinite. Singularities are like fixed points: they can be stable or unstable, they can change stability, and they can be created or destroyed in bifurcations.
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Unformatted text preview: Hint for 4.3.7: For this problem, the bifurcations occur at μ = 1 , , and -1. Plot phase portraits for μ > 1 , < μ < 1 ,-1 < μ < , and μ <-1 , then sketch a bifurcation diagram that includes both fixed points and singularities (no need to distinguish fixed points from singularities, but be careful to label their stability). What type of bifurcations do you see? Hint for 4.3.8: The bifurcations occur at μ = 1 and -1. Draw phase portraits for μ > 1 ,-1 < μ < 1 , and μ <-1 , then put the fixed points and the singularities into a bifurcation diagram. What do you see?...
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