A few questions concerning the Poincare-Bendixson theorem

Question

I have a few questions concerning the Poincare-Bendixson result:($G\subset \mathbb{R}^2$ open, $f\in C_{loc}^{1-}(G,\mathbb{R}^2)$)

Let $u$ be a solution of $u'=f(u)$ so that $\overline{u(\mathbb{R^+})}\subset G$ compact. Assume $\omega(u)\cap E=\varnothing$, where $E$ is the set of equilibria. Then $\omega(u)=u(\mathbb{R}^+)$, i.e. $u$ is a periodic solution or $\omega(u)$ is a periodic orbit.

Question

• According to https://sites.me.ucsb.edu/~moehlis/APC591/tutorials/tutorial3/node2.html a periodic orbit is a solution for which there exists $0<T<\infty$ so that $u(x+T)=u(x)$ for all $x$. So if I have a periodic orbit why is this not a periodic solution?

By Poincare-Bendixson we either have $\omega(u)=u(\mathbb{R}^+)$ or $\omega(u)=u([0,T])$ where $T$ is the period. But if I understand correctly if we have a periodic orbit it already holds $u([0,T])=u(\mathbb{R}^+)$. It seems that this is false, is there a good example or picture to see this?

• Moreover I don't really get the point of homoclines: I'm imagining it as a loop which I run through once (then the period is infinity) or more than ones and then I have the corresponding period, is this the right picture?

• For me homoclinic orbits are periodic solutions. Depending on how often I run through the loop the omega limit set could contain only one point but also the complete loop, is this correct?

Answer 1

1. A periodic orbit is a periodic solution, but $\omega(u)$ being a periodic orbit does not imply that $u$ is a periodic solution. For a simple example, consider the system

$$ \begin{aligned} \dot{x} &= -y + \frac{1}{2}x(1 - x^2 - y^2) \\ \dot{y} &= x + \frac{1}{2}y(1 - x^2 - y^2). \end{aligned} $$

This corresponds to spirals about the origin with $\dot{r} = r - r^3$ and $\dot{\theta}=r$. If $r(0)>1$, this spirals with decreasing radius and asymptotically converges to the set $r=1$. Since $\dot{\theta} = 1$ here, this is a periodic orbit. If $r(0) = 1$, this is a solution, but for $r(0) > 1$ it is just a periodic orbit in the $\omega$-limit set.

2. You are correct, except that the period of a homocline is always infinity. This is because a homocline necessarily starts and ends at the unstable and stable manifolds coming from a single critical point, so it can never leave once it reaches the stable manifold of the critical point. The loop can only be traversed once and it takes $t$ traveling from $-\infty$ to $\infty$ to do it.

3. The $\omega$-limit set only contains the critical point in the case of a homoclinic orbit. The entire orbit is not in the set since the loop is only completed once. What makes homoclinic orbits interesting is that the $\alpha$ and $\omega$-limit sets are equal.