result guarantees that a compact connected invariant set for a flow is strongly homogeneous under the existence of a field of non-degenerate quadratic forms ${\EuScript{J}}$ defined on a neighborhood of this set such a way that the flow derivative $DX_t$ keeps positive cones $C_+(x) : = \{0\} \cup \{ v \in T_xM ; {\EuScript{J}}(x) v > 0\}$ invariants, i.e., $DX_t(C_+(x)) \subset C_+(X_t(x))$, for all $t>0$, $x \in \Lambda$ and the projected quadratic forms on the normal bundle, with respect to the flow direction, are monotonic functions.

\[mthm:strong-homog-equiv\] A compact invariant set $\Lambda$ for $X \in {\mathfrak{X}^{1}(M)}$ is strongly homogeneous with index ${\operatorname{ind}}$ if and only if there is a neighborhood $U\subset M$ of $\Lambda$ and a continuous field of non-degenerate quadratic forms ${\EuScript{J}}$ on $U$ with fixed index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}$ such that the preperiodic set $P_*(X)$ of $X\vert_{U}$ is strictly ${\EuScript{J}}$-separated and the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on $P_*(X\vert_{U})$.

If $\Lambda$ is the maximal invariant set of a trapping region $U$ and we require, in addition, that the field direction must be inside the non-positive cone, we obtain an equivalence between the existence of such a quadratic forms and singular hyperbolicity, as in [@ArSal2012 Theorem 