Translation Arcs and Stability in Two Dimensions

Abstract

Given a closed orbit $\gamma$ of a system of differential equations in the plane

$$ \dot{x}=X(x),\; \; x\in \mathbb {R}^2, $$

the index of the vector field X around $\gamma$ is one. This classical result has a counterpart in the theory of discrete systems in the plane. Consider the equation

$$ x_{n+1} =h(x_n ),\; \; x_n \in \mathbb {R}^2, $$

where h is an orientation-preserving embedding and assume that there is a recurrent orbit that is not a fixed point. Then there exists a Jordan curve $\gamma$ such that the fixed point index of h around this curve is one. The proof is based on the theory of translation arcs, initiated by Brouwer. These notes are dedicated to discuss some consequences of the above result, specially in stability theory. We will compute the indexes associated to a stable invariant object and show that Lyapunov stability implies persistence (in two dimensions). The invariant sets under consideration will be fixed points, periodic orbits and Cantor sets.

Appendix

1.1 Remarks on Embeddings

Given a continuous and one-to-one map $h:\mathbb {R}^2 \rightarrow \mathbb {R}^2$, the theorem of invariance of the domain implies that h is open. In particular $h(\mathbb {R}^2 )$ is an open subset of $\mathbb {R}^2$.

The map h is orientation-preserving ($h\in \mathscr {E}_+$) if

$$ d(h,G,h(p))=1, $$

where the degree is computed on any open and bounded set $G\subset \mathbb {R}^2$ with $p\in G$. It can be proved that the value of this degree is independent of the choice of G and p. Moreover it can only take the values $+1$ or $-1$. In the second case h is orientation-reversing. Embeddings that are isotopic to the identity lie in $\mathscr {E}_+$. Those isotopic to the symmetry $S(x_1 ,x_2 )=(x_1 ,-x_2 )$ are orientation-reversing. We recall that two embeddings $h_0$ and $h_1$ are isotopic if there exists a continuous map

$$ H:\mathbb {R}^2 \times [0,1]\rightarrow \mathbb {R}^2 ,\; \; (x,\lambda )\mapsto H_{\lambda }(x) $$

such that $H_{\lambda }$ is one-to-one for each $\lambda \in [0,1]$, $H_0 =h_0$ and $H_1 =h_1$.

1.2 Lemmas for the Proof of Theorem 1

Lemma 2

Assume that $\mathscr {U}$ is an open subset of the plane and $h:\mathscr {U}\subset \mathbb {R}^2 \rightarrow \mathbb {R}^2$ is a continuous and one-to-one map with a stable fixed point $p=h(p)$. Then, given any neighborhood $\mathscr {W}$ of p, there exists an open and simply connected set $\omega \subset \mathscr {U}\cap \mathscr {W}$ such that $p\in \omega$ and $h(\omega )\subset \omega$.

Proof

It is extracted from [27], page 185. Let us fix a closed disk $\varDelta$ centered at p and contained in $\mathscr {U}\cap \mathscr {W}$. The stability of the fixed point allows us to find an open and connected set G with $p\in G$ and $h^n (G)\subset \varDelta$ for each $n\ge 0$. The open and connected set

$$ \varOmega :=\bigcup _{n\ge 0}h^n (G) $$

is positively invariant under h and contained in $\varDelta$. At first sight $\varOmega$ could be the searched set, because it is contained in $\mathscr {U}\cap \mathscr {W}$ and $h(\varOmega )\subset \varOmega$. However this set can have holes. Let $\widehat{\varOmega }$ be the smallest simply connected domain containing $\varOmega$. Intuitively this set is constructed by filling the holes in $\varOmega$, a more formal construction can be found in Lemma 2.6 of [22]. Since h is an embedding, $h(\widehat{\varOmega })=\widehat{h(\varOmega )}$ and so we can take $\omega =\widehat{\varOmega }$.

Lemma 3

Assume that $h:\mathbb {R}^2 \rightarrow \mathbb {R}^2$ is a continuous and one-to-one map with a fixed point $p=h(p)$ that is stable but not asymptotically stable. Then, for each neighborhood $\mathscr {W}$ of p there exists a point $q\in \mathscr {W}\setminus \{ p\}$ that is recurrent.

Proof

By a contradiction argument assume that there exists a neighborhood $\mathscr {W}$ of p such that $\mathscr {W}\setminus \{ p\}$ does not contain recurrent points. Since p is stable we can find two closed balls $\beta$ and B centered at p and such that $\beta \subset B\subset \mathscr {W}$ and

$$ h^n (\beta )\subset B\; \; \mathrm{for\; each}\; n\ge 0. $$

We claim that $p\in L_{\omega } (x,h)$ for each $x\in \beta$. Indeed, the positive orbit $\{ h^n (x)\}_{n\ge 0}$ is contained in B and so it is bounded. In consequence the limit set $L_{\omega } (x,h)$ is non-empty, compact and invariant under h. This limit set has to contain a minimal set M. All points y in M must satisfy $L_{\omega } (y,h)=M$ and so they are recurrent. Summing up, we can say that $L_{\omega } (x,h)$ is contained in B and has at least one recurrent point. Since p is the only recurrent point in this ball, the claim $p\in L_{\omega } (x,h)$ has been proved. Now we observe that, due to the stability of p,

$$ L_{\omega } (x,h)=\{ p\}. $$

Then all points in $\beta$ are attracted by p and this point should be asymptotically stable.