Index sums of isolated singular points of positive vector fields (Q1034187)

Let \(\dot x = V(x)\), \(x \in \mathbb R^n\), be a smooth \textit{positive} vector field, i.e., the positive orthant \(\mathbb R^n_+ = \{ x\in \mathbb R^n : x_i \geq 0\}\) is forward invariant. For given numbers \(\overline{r} > r > 0\) define the set \(D(r,\overline{r}) = \{x\in \mathbb R^n_+ : r \leq \|x\| \leq \overline{r}\}\). The author proves that if \(V(x)\) has only isolated singular points in \(D(r,\overline{r})\) and has no singular points in \(\partial D(r,\overline{r})\), then the sum of indices of the singular points in \(D(r,\overline{r})\) is equal to \(\deg (f_r,K) - \deg (f_{\overline{r}},K)\), where \(\deg (f_r,K)\) is the standard degree of the map \(f_r(x) = V(x)/\|V(x)\|\), \(x \in \mathbb R^n_+ \cap S^{n-1}_r\), at a regular value \(y=f_r(x) \in \mathbb R^n_- \cap S^{n-1}_1\) (here \(S^{n-1}_r\) is the sphere of radius \(r\) with the center at the origin).