Theorem 1 (section 5.1)

Theorem 1 makes use of an integral noted $I_F$, which is called the index of F on the unit sphere. $I_F$ is in fact an integer; in two dimensions, it is a measure of the number of times $F$ winds around the origin. $I_F$ is by no means trivial to compute, unless $F$ has specific, qualitative properties.

Theorem 1   Following the widespread notations used in Berger & Gostiaux (1992), let us write $I_F=\int_{T}(\frac{F}{\parallel F \parallel})^* \sigma$, where T is a hypersurface diffeomorphic to the unit sphere, such that the inside of $T$ is included in the domain of definition $D$ of $F$ and contains all ss of $F$, and where $\sigma$ is a volume form compatible with the canonical orientation of $T$. If $F$ has at least $1+(-1)^{n} I_F$ stable ss, then $J_F$ has a positive circuit.

Proof. In the following, we denote ss of $F$ by $\{x_1, .., x_p\}$. We have the following result from degree theory (Berger & Gostiaux (1992), p 292):

$$\sum_{i=1}^{p}\mathrm{index}_{x_i}F=I_F$$

For any non degenerate ss $x_i$, we have

$$\mathrm{index}_{x_i}F=\mathrm{sgn}(\det J_{F}(x_i))$$

We will denote the set of ss by $R_1 \cup R_2$, where

• $\forall r\in R_1, \mathrm{sgn}(\det J_{F}(r))=(-1)^n$
• $\forall r\in R_2, \mathrm{sgn}(\det J_{F}(r))=(-1)^{n+1}$

Stable ss are in $R_1$, because complex eigenvalues come in pairs, and if there are only complex eigenvalues then the dimension $n$ must be even. We have

$$\mid R_2 \mid (-1)^{n+1} + \mid R_1 \mid (-1)^{n} = I_F$$

$$\mid R_2 \mid = (-1)^{n+1} I_F + \mid R_1 \mid$$ (4)

Thus, if $\mid R_1 \mid > (-1)^{n} I_F$ (which derives directly from the last hypothesis in the theorem), one derives $\mid R_2 \mid > 0$, and $F$ has a ss $s$ verifying $\mathrm{sgn}(\det J_{F}(s))=(-1)^{n+1}$; according to lemma 1, $J_F(s)$ has a positive circuit. This proves the theorem. Note that the existence of a positive circuit is an open condition : a positive circuit can disappear only if one of its elements vanishes; by taking the finite intersection of neighbourhoods in which each element doesn't vanish, one gets a neighbourhood of the ss in which a positive circuit is always present. $\qedsymbol$

This type of result can also be derived on compact manifolds, replacing the computation of the integral $I_F$ by the computation of the Euler characteristic, but it is not clear whether there are examples of dynamical systems on compact manifolds which are relevant to biological systems. The application of the previous theorem involves knowing $I_F$. Whenever possible, it is probably easiest to derive $I_F$ from qualitative observations (for example, when $F$ is inward-pointing, the determination is direct, see below). For more difficult cases, we propose to compute an approximate value; since it is known from degree theory that $I_F$ is an integer, we can actually turn this approximate value into the exact value, provided we have an upper bound on the integration error, which is rather easy to find. Theorem 1 is interesting in that it replaces an extremely complicated computational problem, solving a multi-dimensional system of non-linear equations (to find ss of $F$) and computing eigenvalues of the Jacobian matrices (to find the existence of a real, positive eigenvalue), into a more straightforward problem of numerical analysis: the computation of an approximate integral. However, computing an approximate integral also becomes very difficult when the dimension rises, because of the exponential dependence of the complexity on the dimension of the problem. If $F$ is inward-pointing, then $I_F=(-1)^n$, and Corollary 2 is thus an immediate consequence of Theorem 1: if $F$ is inward-pointing, and if $F$ has at least $2$ stable ss, then $J_F$ has a positive circuit.