Circular Limit Cycle Homework: Find Fixed Points, Amplitude & Period

[ugresearcher]

Homework Statement

Consider the system x''+ax'(x²+x'²-1)+x = 0, where a>0.
a)Find and classify all the 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.

Homework Equations

y = x'
For fixed points: x'=0, y'=0.

The Attempt at a Solution

Using y=x', the system becomes:
x' = y
y' = -ax'(x²+x'²-1)-x

Using x' = 0 and y' = 0, 1 Fixed point exists at (0,0).

I'm not sure where to go from here. The problem is in Chapter 7, section 1 of "Nonlinear Dynamics and Chaos" by Strogatz.

 

[lippyka]

I've tried looking at the solutions to similar problems, but I'm not familiar enough with the material to understand it. Could someone help me out?