Convex and Concave Differential Systems

We consider a system x'=f(t ,x) of n
first order differential equations, where all
coordinate functions are weakly convex
(or weakly concave ) in x. We have
investigated how the closed solutions
behave in subsets (of suitable form), in
which the off-diagonal entries in the
Jacobi matrix have fixed sign. The
investigations have shown that it is
possible to generalize an earlier
(published) result in the case n=1 on
the number of closed solutions. Furthermore,
we have found some geometrical and
topological properties of the set of
initial points for closed solutions in a
subset of the mentioned type. These
results are in particular interesting in
the cooperative (or the competitive) case.