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527ass7 - them Does this con±rm your analysis above(it...

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642:527 ASSIGNMENT 7 FALL 2007 Multple-page homework must be STAPLED when handed in. Turn in starred problems Tuesday 10/23/2007. Section 7.4: 2 (a)*, (b)*, (k)* In each of these problems, do the following: Find all singular points; Obtain the linearized system near each singular point and classify the origin of that system (as a saddle point, unstable or stable node, unstable or stable focus, or center). If the origin is a saddle point or a node, obtain the special straight-line trajectories. Sketch the phase plane of the linearized system. Discuss the nature of the singularity in the true system to the extent possible; in particular, discuss whether or not you can conclude that the singularity is of the same type as the linearized system. (See the discussion on page 362 of the Hartman- Grobman theorem). For problems 2(b) and 2(k) use Maple (or some other program) to sketch enough of the phase plane to show all the singular points and the behavior of trajectories near
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Unformatted text preview: them. Does this con±rm your analysis above (it should)? Does it answer questions that you could not answer there? • Here is Maple code for problem 2(b); you just have to enter it. It graphs six trajectories by giving six sets of inital conditions. You should ±nd it fairly easy to modify this code to produce the plot for problem 2(k). You may want to change the ranges of the variables and to pick other initial points (you will probably need more than six trajectories to get a good picture of the behavior). Experiment until you get a nice plot. > with(DEtools); > phaseportrait([diff(x(t),t)=1-y(t)^2,diff(y(t),t)=1-x(t)], [x(t),y(t)],t=-10. .10, [[x(0)=1,y(0)=0], [x(0)=1,y(0)=0.75], [x(0)=1.5,y(0)=1], [x(0)=0.5,y(0)=1], [x(0)=1,y(0)=1.3], [x(0)=1,y(0)=-0.5]], x=-1. .3, y=-2. .2,stepsize=.05,linecolor=RED); 1...
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  • Fall '07
  • Technological singularity, saddle point, The Singularity Is Near, phase plane, special straight-line trajectories

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