Trajectory Poisson-stable

Quasiperiodic trajectories are a special case of Poisson-stable trajectories. The latter plays one of the leading roles in the theory of dynamical systems as they form a large class of center motions in the sense of Birkhoff (Sec. 7.2). Birkhoff had partitioned the Poisson-stable trajectories into a number of subclasses. This classification is schematically presented in Sec. 7.3. Having chosen this scheme as his base, as early as in the thirties, Andronov had undertaken an attempt to collect and correlate all known types of dynamical motions with those observable from physical experiments. Since his arguments were based on the notion of stability in the sense of Lyapunov for an individual trajectory, Andronov had soon come to the conclusion that all possible Lyapunov-stable trajectories are exhausted by equilibrium states, periodic orbits and almost-periodic trajectories (these are quasiperiodic and limit-quasiperiodic motions in the finite-dimensional case). 

All other Poisson-stable trajectories are unstable in the sense of Lyapunov. How can such trajectories be of any use in dynamics The answer was found nearly 30 years later. For the first time, the significance of a stable limit set consisting of individually unstable trajectories for explaining the complex and chaotic behavior of nonlinear dynamical processes was recognized by Lorenz in 1963 [87]. 

In the rough case an analysis of the structure of such a limit set (called a quasiminimal set, which is defined as the closure of an unclosed Poisson-stable trajectory) may be performed using Pugh s closing lemma. The main conclusion that follows from this analysis (see Sec. 7.3) is that periodic orbits are dense in a rough quasiminimal set. In particular, we will see that the number of periodic orbits is infinite. Systems possessing such limit sets are called systems with complex dynamics. 

The Poincare-Bendixson theory is also applicable for systems on a cylinder, as well as on a two-dimensional sphere. As for other compact surfaces like tori, pretzels (spheres with a handle) etc., there may exist vector fields that possess, besides equilibria and limit cycles, unclosed Poisson-stable trajectories as well. 

Rough systems are also dense in the space of systems on two-dimensional orientable compact surfaces for which the necessary and sufficient conditions of roughness are analogous to those in the Andronov-Pontryagin theorem. The theory of such systems was developed by Peixoto [107]. The key element in this theory proves the absence of unclosed Poisson-stable trajectories in rough systems (they may be eliminated by a rotation of the vector field). 

Theorem 7.2. (BirkhofF) The Poisson-stable trajectories are dense everywhere in the set of center motions. 

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... 

The Poisson-stable trajectories may be sub-divided into two kinds depending on whether the sequence Tfc(e) of Poincare return times of a P-trajectory to its -neighborhood is bounded or not. Birkhoff named the trajectories of the first kind recurrent trajectories. Such a trajectory is remarkable because regardless of the choice of the initial point, given e > 0 the whole trajectory lies in an -neighborhood of the segment of the trajectory corresponding to a time interval L(e). Obviously, equilibrium states and periodic orbits are the closed recurrent trajectories. 

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... 

In the case of recurrent trajectories, there are certain statistics in Poincare return times which are weaker than that characterizing genuine Poisson-stable trajectories. Nevertheless, there is a particular sub-class of recurrent trajectories which is interesting in nonlinear dynamics. This is the class of the so-called almost-periodic motions. The remarkable feature which reveals the origin of these trajectories is that each component of an almost-periodic motion is an almost-periodic function (whose analytical properties are well studied, see for example [49, 66, 84]). 

Theorem 7.6. (Markov) If a Poisson-stable trajectory is uniformly stable in the sense of Lyapunov then it is almost-periodic. 

This result shows that an individual trajectory cannot give an adequate image of chaotic oscillations. Looking ahead we note that all imclosed Poisson-stable trajectories in structurally stable systems are, in fact, unstable, or more precisely, of the saddle type. 

N is an attracting quasiminimal set which contains two -stable separatrices Fi and F2, one -orbit in W 0) and a continuum of unclosed Poisson-stable trajectories. 

The central sub-class of non-wandering points are points which are stable in the sense of Poisson. The main feature of a Poisson-stable point is not only the recurrence of its neighborhood but the recurrence of the trajectory itself. The definition of Poisson-stable points below is different in some ways but equivalent to the definition given in Chap. 1. 

It is important to distinguish the P , P and P-stable trajectories from each other. Indeed, consider the example from Sec. 1.2 of a system on a two-dimensional torus which possesses an equilibrium state with a P -trajectory which is a-limiting to the equilibrium state and a P -trajectory which is cj-limiting to it all other trajectories on the torus are Poisson-stable, and cover it densely. 

So, one can see that the original system with a Poisson-stable imclosed trajectory will possess infinitely many periodic orbits (p t, xjb), where (/ (0, Xk) = Xk k = 1, 2,...) with periods r, such that Xk xq and Tk +oo as A -> +oo. 

One can see that if Xq is P (P )-stable, its trajectory is P (P )-stable too. Hence, we may generalize the notion of the Poisson stability over semitrajectories and whole trajectories. 

Since each point on a P-trajectory is non-wandering, this result is also valid for points stable in the sense of Poisson. The closing lenuna implies the following meaningful corollary a rough system with a P-trajectory possesses infinitely many periodic orbits. 

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