Poincare return time

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. 

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]). 

Conclusions, (i) For irreversible emission to occur it is necessary that the frequencies kn of the bath oscillators are dense.9 0 If there had been only a few oscillators the energy would shuttle back and forth between them and the main oscillator. The return time (or Poincare period ) would be of the order of the reciprocal of the distance between the kn. 

To determine Tsiow and Tjast in our numerical simulations we have calculated the mean return times of the trajectory to two appropriately chosen Poincare sections... 

C3.6.1(a )), from right to left. Suppose that at time the trajectory intersects this Poincare surface at a point (c (tg), C3 (Sq)), at time it makes its next or so-called first reium to the surface at point (c (tj), c 3 (t )). This process continues for times t, .. the difference being the period of the th first-return trajectory segment. The... 

This second point is quite an interesting one, for there is a theorem known as the Poincare recurrence theorem which states that an isolated system (like our molecule left to itself) will in the course of time return to any of its previous states (e.g. the initial state), no matter how improbable that state may be. This recurrence can be observed with very small molecules but not with polyatomic molecules, because in the latter there are far too many levels of the final state the recurrence time is then far longer than any practicable observation time. 

A Poincare map is established by cutting across the trajectories in a certain region in the phase space, say with dimension n, with a surface that is one dimension less than the dimension of the phase space, n — 1. One such cut is also shown in Fig. 1. The equation that produces the return to the crossing the next time is a discrete evolution equation and is called the Poincare map. The dynamics of the continuous system that creates the Poincare map can be analyzed by the discrete equation. Therefore, chaotic behavior of the Poincare map can be used to identify chaos in the continuous system. For example, for certain parameters, the Henon discrete evolution equation is the Poincare map for the Lorenz systems. 

Since the recurrent time of a nearby orbit to a cross-section is about uj p) (see the last section), it follows that the period of the orbits of the flow which corresponds to the fixed points of the Poincare map tends to infinity as p - -0 (typically, it is /y/pt ) Before the orbits return to the cross-section, each must make uj p) rotations in a small neighborhood of the just disappeared saddle-node L. Accordingly, the length of these periodic orbits is also increasing to infinity. Thus, Theorem 12.8 gives a positive answer to the following... 

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