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Unformatted text preview: 3* (Glendinning 5.6) Analyze the xed points of the system x = x 2-y-1 , y = ( x-2 ) y . Plot the trajectories. 4 Show that the origin is a xed point of the system x = ( x 2 + y 2 ) x 2 x 2 + y 2 ( 1-ax 2-ay 2 )-y x 2 + 2 y 2 , y = ( x 2 + y 2 ) y 2 x 2 + y 2 ( 1-ax 2-ay 2 )+ x x 2 + 2 y 2 , and discuss its nature. For a > 0 show that r > 0 for small r and r < 0 for large r . The Poincare Bendixson theorem shows that there is then a periodic orbit; explain why this makes sense from a graphical perspective. Now transform to polar coordinates, nd the periodic orbit and solve for r ( ) in the general case. 1...
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- Winter '08