Poincaré-Bendixson theorem in $\mathbb{R}^1$

I have a question about a problem in Teschl's ODE&DS book.

Consider a continuous dynamical system induced by an autonomous ODE $$\dot{x} = f(x)$$, $$f \in C^1(M, \mathbb{R}^2)$$ where $$M$$ is an open subset of $$\mathbb{R}^2$$.

Poincaré-Bendixson theorem says that if the $$\omega$$-limit set $$\omega_{\sigma}(x)$$ of $$x \in M$$ is nonempty compact connected and contains only finitely many fixed points, then $$\omega_{\sigma}(x)$$ is a fixed orbit, or a regular periodic orbit, or consists of finitely many fixed points and non-closed orbits.

• There cannot be any regular periodic orbit since the orbit must pass a fixed point by the mean value theorem, which seems absurd.

• If there are finitely many fixed points of $$f$$, then I guess that $$\omega_{\sigma}(x)$$ is a subset of the collection of fixed points, with size $$1$$ (i.e. a singleton), and non-closed orbits $$\gamma (y)$$ with $$\omega_{\sigma}(y)$$ is a subset of the collection of fixed points, with size $$1$$.

• If there is not a fixed point of $$f$$, then I guess that $$\omega_{\sigma}(x)$$ is an empty set $$\gamma_{\sigma}(x)$$.

I'm not sure about what the author wants in this question. What does a 'Poincaré-Bendixson theorem' in $$\mathbb{R}^1$$ mean?

edited:

Your last question asks what the author means by a Poincaré–Bendixson theorem on the real line—but you're on the right track: it's asking you to categorise $$\omega$$-limit sets, whenever they reasonably exist.

Notice how, given your $$\mathcal C^1$$ ODE on some open interval $$U \subset \mathbb R$$,

$$\dot x = f(x),$$

treating $$\mathbb R = \mathbb R \times \{0\}$$, you can always enlarge this to an ODE on the plane: i.e.,

$$\ \dot x = f(x), \quad \dot y = 0.$$

So P-B's theorem in 2D certainly applies here (you can project down to 1D again as you like): an omega limit set must be a homoclinic orbit, a fixed point, or a periodic loop.

Further, as you almost-correctly hinted at, only one of these three are possible on the real line (e.g. periodic orbits are ruled out by the MVT and the uniqueness-of-solutions, as you rightly pointed out). Indeed, some of the restrictions to the 2D P.-B. can be relaxed.

(Ruling out homoclinic tangencies is a little trickier. To start: how can your orbit get close to all of the points on the homoclinic tangency? Only by being on it. . .)

However, it is impossible to have more than one fixed point in the omega limit set without there being a homoclinic tangency, since the statement of P-B does not have this as an option (in the 2D version).

In sum, your statement would read something like this: given a $$\mathcal C^1$$ ODE on an open set $$U$$ in $$\mathbb R$$, then if $$x \in U$$ and $$\omega(x)$$ is non-empty, then it consists of only one fixed point.

(Yes, that may have been what you were trying to say in the above.)

I suppose that the idea in the question is that you should digest the definition of what it means to be an $$\omega$$-limit and conclude that in one dimension the $$\omega$$-limit must be trivial, either empty or a point. Hint: Let $$x(t), t\geq 0$$ be the trajectory of $$x$$. If $$y,z \in \omega(x)$$ then there are time sequences $$(t_n)_n, (t'_n)_n$$ with $$t_n, t'_n\rightarrow +\infty$$ as $$n\rightarrow +\infty$$ such that $$\lim_n x(t_n) = y \ \ \ \mbox{and} \ \ \ \lim_n x(t'_n) = z.$$ Show that in one dimension this is impossible unless $$y=z$$. It is very simple, and using the 2D-Poincare-Bendixson in this context would be like using a sledgehammer to crack a nut.