Butler-McGehee theorem

Let p = x(0) be an arbitrary initial point with Jr,(0) > 0. Then the initial data do not belong to either stable manifold. Hence w p) is not equal to either Eq or E2, but it does lie on E = 0. Since it is invariant and since every solution of (5.2) on E = 0 converges to an equilibrium, u(p) contains an equilibrium. By the Butler-McGehee theorem, Eq uIp) since M (Eq) is unbounded. If o p) contains E2, then o p) also contains either Eq or an unbounded orbit, again by the Butler-McGehee theorem (see Figure 5.2). Since this is impossible, E must be in 0 p). However, is a local attractor, so u p) = Ey This completes the proof. ... 

Proof. Note that M (Eq), the stable manifold of Eq, is either the p axis if El exists or the x -p plane if Ei does not exist. The manifold M E2) is the X2 p plane less the p axis if E exists, M (Ei) is the Xi p plane less the p axis. Since (Xi(0), X2(0), p(0)) does not belong to any of these stable manifolds, its omega limit set (denoted by w) cannot be any of the three rest points. Moreover, w cannot contain any of these rest points by the Butler-McGehee theorem (see Chapter 1). (By arguments that we have used several times before, if w did then it would have to contain Eq or an unbounded orbit.) If w contains a point of the boundary of then, by the invariance of w, it must contain one of the rest points Eq,Ei,E2 or an unbounded trajectory. Since none of these alternatives are possible, CO must lie in the interior of the positive cone. This completes the proof. 

Suppose E ew(q). Since q M Ec), it follows that w(q) E. The Butler-McGehee theorem implies that w(q) contains a point r of M (Ec) distinct from E. By Proposition 6.4, either r> E or r[Pg.97]

Theorem (Butler-McGehee). Suppose that P is a hyperbolic rest point of (3.1) which is in w(x), the omega limit set of but is not the en-... 

The stable and unstable sets correspond to the stable and unstable manifolds introduced for rest points and periodic orbits in Chapter 1. Unfortunately, if the attractors are more complex than rest points or periodic orbits, the question of the existence of stable and unstable manifolds becomes a difficult topological problem. In the applications that follow, these more complicated attractors do not appear, so one can simply deal with the stable manifold theorem. The Butler-McGehee lemma (used in Chapter 1) played a critical role in the first uses of persistence. The following lemma is a generalization of this work. It can be found (with slightly different hypotheses) in [BW], [DRS], and [HaW]. (In particular, the local compactness is not needed if a stronger condition - asymptotic smoothness - is placed on the semidynamical system.)... 

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