. Consider the van der Pol systemx= y, y= x + (1

Chapter 9, Problem 17

(choose chapter or problem)

. Consider the van der Pol systemx= y, y= x + (1 x2)y,where now we allow the parameter to be any real number.(a) Show that the origin is the only critical point. Determine its type, its stability property,and how these depend on .(b) Let = 1; draw a phase portrait, and conclude that there is a periodic solution thatsurrounds the origin. Observe that this periodic solution is unstable. Compare your plotwith Figure 9.7.4.(c) Draw a phase portrait for a few other negative values of . Describe how the shapeof the periodic solution changes with .(d) Consider small positive or negative values of . By drawing phase portraits, determinehow the periodic solution changes as 0. Compare the behavior of the van der Polsystem as increases through zero with the behavior of the system in 16. 18 and 19 extend the consideration of the RosenzweigMacArthur predatorprey model introduced in of Section 9.5.

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