Determine the stability of the limit cycle

Question

Question 1: (A circular limit cycle) Consider x¨+ ax·(x² +x^·2 -1) + x = 0, where a > 0.

i) Find and classify all the fixed points.

ii) Show that the system has a circular limit cycle, and find its amplitude and period.

iii) Determine the stability of the limit cycle.

iv) Give an argument which shows that the limit cycle is unique, i.e., there are no other periodic trajectories.

Question 2: Sketch the phase portrait for each of the following systems. (As usual, r, θ denote polar coordinates.)

7.1.1  r· = r³ - 4r, θ = 1    7.1.2 r· = r(1 - r² )(9 - r²), θ·=1

7.1.3 r· = r(1- r² )(4 - r²), θ· = 2 - r² 7.1.4 r·  = r sin r, θ·= 1