The orbit structure of uniformly positive cyclic systems

Abstract

In this paper, we apply the concepts introduced in the immediately preceding paper [3] to obtain further insight into the orbit structure of third-order positive cyclic systems studied in [I, 21. For this purpose, we first extend (in Section 2) to autonomous families of differential systems some old ideas of G. D. Birkhoff’s theory of dynamical systems [j]. In particular, we extend to autonomous families the definition of minimal sets and observe that they always exist on compact manifolds. We nest make (in Section 3) some general observations, which are immediate corollaries of our earlier work, about uniformly positive cyclic systems of arbitrary order. We then consider autonomous families of third-order uniformly positive cyclic systems and examine the projections on the unit sphere s” of their orbits (i.e., phase space trajectories). We determine on S2 the elliptic, parabolic, and hyperbolic points and the minimal sets of these families. We describe (in Theorem 9) aperiodic solution which (projectively) bounds an “equatorial belt” containing all the oscillatory solutions. Finally, we show that the asymptotic qualitative behavior (as t 4 &COO) can be described quite well by reference to this belt and an orthogonal “polar cap” containing all the positwe solutions.