Determining if $u''+(u^2+u'{^2}-1)u'+u=0$ has a limit cycle

Question

I am trying to determine if the ODE $$u''+(u^2+u'{^2}-1)u'+u=0,$$ has a limit cycle.

Converting this ODE into a system of first-order ODES, $$\underline{x}=\begin{pmatrix} x \\ y \end{pmatrix}, \ \ \underline{x}'=\begin{pmatrix} y \\ -y(x^2+y^2-1)-x \end{pmatrix}.$$ Using polar coordinates, we can show that $$r'=-r\sin^2(\theta)(r^2-1).$$ Clearly, $r'=0$ when $r=0,1$ and when $\theta=n\pi\ $ for $\ n\in\mathbb{Z}.$ I have been instructed that $r=0$ is a fixed point, my question is why? From my understanding, a fixed point was such that $r'=0$ and $\theta'=0$, but the equation for $\theta'$ is $$\theta'=-1-\sin(\theta)\cos(\theta)(r^2-1).$$ For $r=0$, $\theta'\neq 0.$

Answer 1

The polar coordinates are $u=r\cosθ$, $u'=r\sinθ$. Thus if $r=0$ and stays there, you get $(u,u')=(0,0)$ for all times, a fixed, stationary or equilibrium point.

Answer 2

Let $C_1$ and $C_2$ be two closed cycles $r=\frac12$ and $r=\frac32$, respectively. Let $R$ be the region between $C_1$ and $C_2$. At $C_1$, $$ r'\bigg|_{r=\frac12}=-r\sin^2(\theta)(r^2-1)\bigg|_{r=\frac12}\ge0 $$ which means that $r(t)$ points toward the interior of $R$ at each point of $C_1$. Similarly, $r(t)$ points toward the interior of $R$ at each point of $C_2$. By the Poincare-Bendixson Theorem, there is a limit cycle lying inside of $R$. Actually, the limit cycle is $r=1$ since it is an equilibrium point.