Whenever I say ... dot I mean the derivative of ...

I have to show that the system of Ordinary Diff. Eqns

x (dot) = y + x* f(r) / r

y (dot) = -x + y * f(r)/r

r=sqrt(x^2+y^2)

has limit cycles and the limit cycles have radius which correspond to the zeros of f(r)

Also I have to find the direction of motion on the limit cycles.

My workings:

I first found the nullclines.

C(x): y + x* f(r) / r =0 and C(y): -x + y * f(r)/r =0 implies that x=y*f(r) / r

Substituting into C(x) we get that the system has an equilibrium point at

(y * f(r) / r, 0)

Then we consider polar coordinates. Using r^2=x^2 + y^2

Then differentiate to obtain that r dot = (f(r) / r^2 ) * (2x^2 + 2y^2)

Now if 2x^2 + 2y^2 = 0 => r dot = 0

if 2x^2 + 2y^2 > 0 => r dot >0

Now I have to find C1 and C2 s.t they bound a region R that contains no equilibrium points and trap all trajectories so I can use Poincaré–Bendixson theorem to show that there exists a limit cycle.

Any help would be greatly appreciated!

Thank you!