Non-wandering set

Theorem 7.8. (Pugh) Arbitrarily close in -sense) to any smooth flow, there exists a flow for which the periodic orbits are dense everywhere in the non-wandering set" ... 

The validity of the C -version of this theorem with r > 2 remains unknown up-to-date. Strictly speaking — in the non-wandering set minus the equilibrium states. 

Axiom 1. The non-wandering set 11 consists of a finite number of orbits. 

Another example is a family of two-dimensional C -smooth diffeomor-phisms whose non-wandering set does not change until the boundary of Morse-Smale diffeomorphisms is reached. The situation is illustrated in Fig. 8.2.3. The two fixed points 0 and O2 have positive multipliers, and Wq contacts Wq along a heteroclinic trajectory, and so do Wq and. This example... 

There is no doubt that some subtle aspects of the behavior of homoclinic and heteroclinic trajectories might not be important for nonlinear dynamics since they refiect only fine nuances of the transient process. On the other hand, when we deal with non-wandering trajectories, such as near a homoclinic loop to a saddle-focus with i/ < 1, the associated fi-moduli (i.e. the topological invariants on the non-wandering set) will be of primary importance because they may be employed as parameters governing the bifurcations see [62, 63]. 

Since the set of wandering points is open, its complement, which is the set of non-wandering points, is closed. We will denote it by Afi. Let us show that it is not empty under our assumptions. First of all, notice that the set of (j-limit points of any semi-trajectory is non-empty. This follows from the compactness of G,... 

Let us return to the set A4i of non-wandering points. We have established that it is non-empty, closed and invariant (consists of whole trajectories). The set Ail may be regarded as the phase space of a dynamical system, and therefore one may repeat the procedure and construct the set A 2 consisting of non-wandering points in A i. Clearly, Ai2 Q Ai. Just like Ai, the set AA2... 

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. 

It may be proved that the relation < defines a partial order on the set of non-wandering orbits of a Morse-Smale system. An important result is ... 

The closure of an unclosed Poisson-stable trajectory whose return times are unbounded for some e > 0 is called a quasiminimal set. A quasiminimal set contains, besides Poisson-stable trajectories which are dense everywhere in it, some other invariant and closed subsets. These may be equilibrium states, periodic orbits, non-resonant invariant tori, other minimal sets, homoclinic and heteroclinic orbits, etc., among which a P-trajectory is wandering. This gives a clue to why the recurrent times of the non-trivial unclosed P-trajectory are unbounded. Furthermore, this also points out that Poisson-stable trajectories of a quasiminimal set, due to their unpredictable behavior in time, are of... 

---