402_502_W11_hw6 - occurs, and whether it was supercritical...

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AMATH 402/502 Homework DUE at time and place posted on website 1 Problem 7.3.1 2 Problem 7.3.4 3 Problem 8.1.6 For (c), choose values of μ that demonstrate di±erent behavior on “all sides” of the bifurcation(s). That is, just choose the appropriate number of μ values and make phase portraits for these. 4 Problem 8.1.12 (parts (a-d) only). 5 The van der Pol oscillator. Consider: d 2 x dt 2 + ( x 2 - v ) dx dt + x = 0 where v is a parameter that can take any real value. a Find all ²xed points, and the Jacobian evaluated as these ²xed point(s). b Classify the ²xed point(s), for all possible values of v . c Identify all value(s) of v where a Hopf bifurcation could occur. d Say you were going to use pplane to ²gure out whether or not a Hopf bifurcation actually
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Unformatted text preview: occurs, and whether it was supercritical or subcritical if it did? What exactly would you do? Please describe, including what values of v you would use, and what you would have to use at those values of v to conclude that a supercritical or subcritical Hopf had occurred. Be as precise as you can. e (ungraded). Carry out your program in [d]. What do you nd? Ungraded bonus: keep changing v , well beyond any bifurcation values. Does your limit cycle persist? The homework will be graded statistically. Late homework is not accepted. Your homework should be neat and readable (the TA is allowed to subtract points for presentation)....
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