I am trying to prove that the following dynamical system has a limit cycle by rewriting it in polar coordinates
$\dot{x} = x-y-x^2(x+2y)-xy^2$
$\dot{y}= x+y+x^2(x-y)-y^2(x+y)$
Using the identities
$r\dot{r}= x\dot{x}+ y\dot{y}$
$\dot{\theta}=\frac{x\dot{y}-y\dot{x}}{r^2}$
I have obtained
$\dot{r}=r-r^3(cos^4\theta +sin^4\theta)-r^3(cos\theta sin\theta(cos2\theta-cos\theta sin\theta))$
$\dot{\theta}=2+rcos^3\theta-cos\theta sin\theta$
But how can I then show that a limit cycle exists?
Best Answer
Transform the radius dynamic using $$\cos^4 θ+\sin^4 θ=1-2\sin^2θ\cos^2θ$$ and $$2\sinθ\cosθ=\sin(2θ),$$ as well as Cauchy-Schwarz $$ |a\cosθ+b\sinθ|\le\sqrt{a^2+b^2} $$ to get $$ \dot r=r-r^3(1+\tfrac12\sin(2θ)(\cos(2θ)-\tfrac32\sin(2θ)) =r-r^3(1+\tfrac14\sin(4θ)-\tfrac38(1-\cos(4θ))). $$ From this result the angle-free bounds $$ r-\frac{5+\sqrt{13}}8r^3\le \dot r\le r-\tfrac{5-\sqrt{13}}8r^3 $$ So for $r^2<\frac8{5+\sqrt{13}}$ the radius is always growing, for $r^2>\tfrac8{5-\sqrt{13}}=\frac{2(5+\sqrt{13})}3$ the radius is always falling, the annulus in-between is a trapping region.
One would have to correct the angle dynamic formula, as currently it does not have the correct degrees. Then make sure that there are no stationary points for the angle inside the annulus to exclude overall stationary points.