Question: a) For the system x = y ( y x ) , y = x 2 a 2 , a > 0 , show that

a) For the system x=y(yx),y=x2a2,a>0, show that the index for any closed curve enclosing all four points (a, 0), (-a, 0), (a, a) and (-a, -a) is 0.

b) Hence show that the system x=y(yx),y=x2, has no limit cycle.

These are problems from the course "Nonlinear Ordinary Differential Equations". Please provide two perfect proofs for each problem.

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