Some Applications of the Poincaré–Bendixson Theorem

Abstract

We consider a ${\mathcal {C}}^1$ vector field X defined on an open subset U of the plane with compact closure. If X has no singular points and if U is simply connected, a weak version of the Poincaré–Bendixson theorem says that the limit sets of X in U are empty but that one can define non empty extended limit sets contained in the boundary of U. We give an elementary proof of this result, independent of the classical Poincaré–Bendixson theorem. A trapping triangle ${\mathcal {T}}$ based at p, for a ${\mathcal {C}}^1$ vector field X defined on an open subset ${\mathcal {U}}$ of the plane, is a topological triangle with a corner at a point p located on the boundary $\partial {\mathcal {U}}$ and a good control of the tranversality of X along the sides. The principal application of the weak Poincaré–Bendixson theorem is that a trapping triangle at p contains a separatrix converging toward the point p. This does not depend on the properties of X along $\partial {\mathcal {U}}$. For instance, X could be non differentiable at p, as in the example presented in the last section.

Appendix: The Jordan–Schoenflies Theorem in class \({\mathcal {C}}^1\)

We want to give a sketch of the proof for the following restricted version of the Jordan-Schoenflies Theorem, the only version used in the present article. The support of a diffeomorphism H of ${{\mathbb {R}}}^2$ is the closure of the set: $\{m\in {{\mathbb {R}}}^2 |\ m\not = H(m)\}$.

Theorem 5.1

(Jordan-Schoenflies Theorem in class ${\mathcal {C}}^1$) Let $\Gamma$ be a ${\mathcal {C}}^1$ closed regular curve in ${{\mathbb {R}}}^2.$ There exists a ${\mathcal {C}}^1$ diffeomorphism H of ${{\mathbb {R}}}^2,$ with compact support, sending the trigonometric circle onto $\Gamma .$

Proof

Let (x, y) be Cartesian coordinates in the plane ${{\mathbb {R}}}^2$. We choose a tubular neighborhood T of $\Gamma$ with a ${\mathcal {C}}^1$ trivialization $T\cong S^1\times [-1,+1]$ such that $S^1\times \{0\}$ corresponds to $\Gamma$. The segments $\{\theta \}\times [-1,+1]$ give a ${\mathcal {C}}^1$ normal fibration ${\mathcal {N}}$ along $\Gamma .$ A first step is to approach $\Gamma$ by a ${\mathcal {C}}^\infty$ closed curve $\Gamma _1$ in the ${\mathcal {C}}^1$ topology. In a second step $\Gamma _1$ can be approached in the ${\mathcal {C}}^\infty$ topology by a smooth closed $\Gamma _2$ in generic position in relation with the foliation ${\mathcal {F}}$ by the horizontal lines $\{y=\mathrm{Const.}\}$, meaning that all contact points of $\Gamma _2$ with ${\mathcal {F}}$ are quadratic and located on different leaves (see Fig. 10). We can choose $\Gamma _2$ sufficiently near $\Gamma$ in order that $\Gamma _2$ is inside the interior of T and transverse to ${\mathcal {N}}$. Then, $\Gamma _2$ is given in the trivialization, by the graph of a map from $S^1$ to $]-1,1[$ and it is easy to construct a ${\mathcal {C}}^1$ diffeomorphism of ${{\mathbb {R}}}^2$, with support in T, sending each fiber of ${\mathcal {N}}$ onto itself and sending $\Gamma$ onto $\Gamma _2$ (see Fig. 11).

Fig. 10

A curve $\Gamma _2$

Fig. 11

Smoothing $\Gamma$

We now consider the smooth curve $\Gamma _2.$ The position of $\Gamma _2$ with respect to the horizontal foliation may be rather complicated, with a lot of horizontal contact points, as it is suggested in Fig. 10. In the rest of the proof we explain how to simplify this by means of a diffeomorphism of ${{\mathbb {R}}}^2$ with compact support.

If there are just two such horizontal contact points, a maximum and a minimum for the y-function, we can displace $\Gamma _2$ by an affine map such that the maximum is on the line $\{y=1\}$ and the minimum on the line $\{y=-1\}.$ It is now very easy to construct a smooth diffeomorphism $G(x,y)=(g(x,y),y)$ sending $\Gamma _0$ onto $\Gamma _2$ (see Fig. 12)

Fig. 12

Curve $\Gamma _2$ with two contact points

If there are more than 3 contact points, it is possible to prove that there is at least a pair of two successive contact points on $\Gamma _2$, a minimum p and a maximum q, in the position illustrated in Fig. 13: there is a disk B such that $\partial B$ is the union of an arc $\Gamma _2(p,p')$ on $\Gamma _2$ containing q in its interior (and no other contact point) and the horizontal segment $[pp']$. Moreover, at p, the complement $\Gamma _2\setminus \Gamma _2(p,p')$ starts outside B. The disk B may contain other parts of $\Gamma _2$ disjoint from $\Gamma _2(p,p')$. Let $\Gamma _B$ be their union. The existence of such a pair (p, q) of contact points is the key point of the proof. It can be obtained by an easy recurrence argument on the number of contact points. We will not elaborate further on it.

We now consider such a pair (p, q) and explain how to eliminate it. We proceed in two sub-steps. First, we choose a disk W disjoint from $\Gamma _2(p,p')$, and such that $\Gamma _B$ is inside $W\cap B$. Then, we push $\Gamma _B$ outside B by a smooth diffeomorphism, with support in W which, in a neighborhood of $\Gamma _B$, sends horizontal intervals into horizontal intervals located outside B (see Fig. 13).

Fig. 13

Cleanning out a disk B

We obtain a new curve $\Gamma _3$, in generic position and coinciding with $\Gamma _2$ in a neighborhood of $\Gamma _2(p,p')$, with the same number of horizontal contact points as $\Gamma _2.$ But now the same disk B is associated to $\Gamma _3$ and does not contain other parts of $\Gamma _3$ than $\Gamma _2(p,p')$. It is now easy to construct a diffeomorphism, with support in a compact neighborhood of B, which pushes the arc $\Gamma _2(p,p')$ downward outside B, in order to eliminate the pair (p, q) without modifying the other contact points nor creating new ones (see Fig. 14)

Fig. 14

Elimination of the pair (p, q)

One can apply this argument by recurrence to finish with a curve which has just two contact points. Finally, we have obtained a sucession of diffeomorphisms of ${{\mathbb {R}}}^2,$ with compact support and of class at least ${\mathcal {C}}^1$, whose composition (of them or their inverse) sends the trigomometric circle onto the given initial closed curve $\Gamma$. $\square$