402_502_W11_hw5 - 3 Problem 6.5.4 4 Problem 6.5.8 5 Problem...

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AMATH 402/502 Homework DUE at time and place posted on website 1 Problem 6.3.1 2 Problem 6.3.10, (a-c) only. For the context of part (c) of this problem, we will say that a fixed point is a saddle – regardless of linearization – if you can (1) find a direction along which trajectories approach the fixed point, and (2) argue that NOT all trajectories that start nearby the fixed point approach it (i.e., that there is a “direction” for which trajectories escape small neighborhoods). Looking at the ˙ y = 0 nullcline should be useful for (2). Please give a simple, reasonable geometrical picture for this based on your phase plot and some tangent arrows, not a full epsilon-delta proof.
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Unformatted text preview: 3 Problem 6.5.4 4 Problem 6.5.8 5 Problem 6.5.12 6 Problem 7.1.5 7 Problem 7.2.6 (a) 8 Problem 7.2.9 (a,b). Although there is more than one way to do this, please use the approach of problem 7.2.6, noting whether or not it succeeds in each case. Ignore the part of the problem statement that says “go on to the next problem.” If it is not a gradient dynamical system, just show your work and you’ll be done with the problem. 9 Problem 7.2.10 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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