Consider the second order differential equation + ai (x + x - 1) + x =...

 

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Consider the second order differential equation ï+ ai (x² + x² - 1) + x = 0 where a > 0. (a) Find and classify all fixed points. (b) Show that the system has a circular limit cycle and find its amplitude and period. (c) Determine the stability of the limit cycle. (d) Give an argument that shows that the limit cycle is unique, i.e., there are no other periodic trajectories. Consider the second order differential equation ï+ ai (x² + x² - 1) + x = 0 where a > 0. (a) Find and classify all fixed points. (b) Show that the system has a circular limit cycle and find its amplitude and period. (c) Determine the stability of the limit cycle. (d) Give an argument that shows that the limit cycle is unique, i.e., there are no other periodic trajectories.