Birkhoff’s theorem

Why is the center so remarkable First, this is the set about which all trajectories of the system linger much longer than elsewhere, most of the time. Second, this center is characterized by Birkhoff s theorem. 

In the preceding sections, we have discussed the set of center motions. In essence, we have found that it is the closure of the set of Poisson-stable trajectories. It does not exclude the case where the latter ones may simply be periodic orbits. But if there is a single Poisson-stable unclosed trajectory, then by virtue of Birkhoff s theorem in Sec. 1.2, there is a continuum of Poisson-stable trajectories. As for the rest of the trajectories in the center, it is known that the set of points which are not Poisson-stable is the union of not more... 

In fact, it can be shown that periodic orbits and equilibrium states are the only non-wandering trajectories of Morse-Smale systems. Axiom 1 excludes the existence of unclosed self-limit (P-stable) trajectories in view of BirkhofF s Theorem 7.2. The existence of homoclinic orbits is prohibited by Theorems 7.9 and 7.11 below. Next, it is not hard to extract from Theorem 7.12 that an u)-limit (a-limit) set of any trajectory of a Morse-Smale system is an equilibrium state or a periodic orbit. 

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