Proof for inward-pointing fields

Steady states are the points of state space at which it is easiest to derive conditions about the Jacobian matrix of the system. Studies of the bifurcations of small-dimension systems often involve an unstable steady state. We follow the same approach for higher dimensions: having shown that certain types of unstable ss provide us with the existence of a positive circuit, we now proceed by showing that, under certain conditions, such an unstable ss can indeed be found, which will prove Thomas's conjecture under those conditions. Theorem 1, which can be found in the appendix, makes use of degree theory and of the value of an integral involving $F$, which can be approximated by numerical methods. It is difficult to state beforehand whether our theorem will be applicable to a given vector field, but we think it should be able to cover a wide variety of systems. However our theorem does apply to all systems such that $F$ is inward-pointing, providing the following corollary:

Corollary 2   If $F$ is inward-pointing, and if $F$ has at least $2$ stable ss, then $J_F$ has a positive circuit.