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(Dealing with a fixed point for which linearization is inconclusive) The goal of this exercise is to sketch the phase portrait for x = xy, y = x2 - y a) Show that the linearization predicts that the origin is a non-isolated fixed point

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(Dealing with a fixed point for which linearization is inconclusive) The goal of this exercise is to sketch the phase portrait for x = xy, y = x² - y a) Show that the linearization predicts that the origin is a non-isolated fixed point. b) Show that the origin is in fact an isolated fixed point. c) Is the origin repelling, attracting, a saddle, or what? Sketch the vector field along the nullclines and at other points in the phase plane. Use this information to sketch the phase portrait. d) Plot a computer-generated phase portrait to check your answer to (c). (Note: This problem can also be solved by a method called center manifold theory, as explained in Wiggins (1990) and Guckenheimer and Holmes (1983).)