$ to the vector $\left(  1,2\right)  $.

The local analysis of the fixed points of $F$ reveals a very symmetric picture. When $\varepsilon>0$ is small ($0<\varepsilon<{\varepsilon}_{0}%
=\frac{1}{9}$ is good enough), $F$ is a small perturbation of the identity, $F(\partial{S})=\partial{S}$, the restriction of $F$ to the boundary of $S$, $\partial{S}$, is a bijection (see section 4 for more details), and the Jacobian determinant of $F$ is never null in the interior of $S$. Therefore, $F$ is invertible on S.

Heteroclinic connections and invariant sets
-------------------------------------------

We focus our attention on the existence of invariant subsets of $S$ for the dynamics of $F$. Additionally, we below prove that $S$ is itself an invariant set for the dynamics of $F$.

Recall that an *heteroclinic* (sometimes called a heteroclinic connection, or heteroclinic orbit) is a path in phase space which joins two different equilibrium points. In the sequel, by *sa-heteroclinic*, *rs-heteroclinic*, and *ra-heteroclinic*, we mean an heteroclinic orbit connecting a saddle point to an attractor, an heteroclinic orbit connecting a repeller to a saddle point, and an heteroclinic orbit connecting a repeller to an attractor, respectively.

Let $F$ be our model map in some set $T$ with two fixed points $p$ and $q$. Let 