Orbitals asymptotic conditions

Every vector in (labelled ip n) or [ 0out)) does represent the asymptote of some actual orbit that is, for every vector [ - in) in H there is a solution U(t) 1 0) of the Schrodinger equation that is asymptotic to the free orbit U t) ipin) as -oc and likewise for every I out) as t -hoo. This result is known as the asymptotic condition. If the potential V r) falls off fast enough (for r —> oc), then for every [ in) in H there is a [ > such that... 

The asymptotic condition guarantees that any ip n) in H is in fact the in asymptote of some actual orbit U t) ip). The actual state xp) of the system at =0 is linearly related to the in asymptote ipin) specifically,... 

The Valley theorem leads to simple conditions for the optimised orbitals near the nuclei. However these conditions are not sufficient to characterize these orbitals one needs in addition to take the asymptotic form of the equations into account. 

For a chemical reaction system, the characteristics of the periodic solutions are uniquely determined by the kinetic constants as well as by the concentrations of the reactants and final products. Starting from the neighborhood of steady state as an initial condition, the system asymptotically attains a closed orbit or limit cycle. Therefore, for long times, the concentrations sustain periodic undamped oscillations. The characteristics of these oscillations are independent of the initial conditions, and the system always approaches the same asymptotic trajectory. Generally, the further a system is in the unstable region, the faster it approaches the limit cycle. 

Proof of Theorem 5.3. Condition (3.4) makes E locally asymptotically stable. By the Poincare-Bendixson theorem, it is necessary only to show that with condition (3.4) there are no limit cycles. Suppose there were a limit cycle. However, there is at most a finite number of limit cycles and each must contain < in its interior. Hence there is a periodic trajectory V that contains no other periodic trajectory in its interior. Intuitively speaking, r is the trajectory closest to the rest point. The constant term in the formula given in Lemma 5.1 is negative. The corollary shows that P is asymptotically stable. This is a contradiction, since the rest point is asymptotically stable - that is, between the two there must be an unstable periodic orbit. ... 

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.)... 

Then x is globally asymptotically stable for all initial conditions, x(r) —> x as r oo. In particular the system has no closed orbits. (For a proof, see Jordan and Smith 1987.)... 

In order to check the universality of this observation, we computed numerically similar synchronization diagrams for larger arrays. The results are shown in Fig. 6.21 for 5 and 10 oscillators. One can recognize the same qualitative trend. Note, that the synchronization considered here corresponds only to the asymptotic coincidence of the variables of the outer lasers, i.e. = Ei,Nn = N. This is not sufficient for the complete synchronization of the whole array. In general, the synchronization of the outer lasers may include some cluster states as well [3, 4], Figure 6.21 has been obtained by computing asymptotic behavior of orbits starting from randomly chosen initial conditions. 

Fig. 6.21. The regions in the f — r] parameter space, for which an orbit started from randomly chosen initial condition is attracted to the synchronization subspace Ms This corresponds to the asymptotic synchronization of the outer lasers, (a) 5 coupled lasers, (b) 10 coupled lasers.

In the limit - 0, y(T) changes much more rapidly than x(t) Except near Q = 0 the vector field (x,y) is everywhere nearly horizontal. The two falling sections of the one-dimensional manifold Q 0 are stable, but the middle section is unstable. (We referred to this fact earlier.) For 0 < 6 < 6 and 6 < 6 < /e find that the steady state is globally asymptotically stable (as -> 0). However, under these conditions the system is excitable in the sense described in Chapter IV (pp. 76f) For 6q < 6 < 6 we find an orbitally as3nmptoti-cally stable periodic solution illustrated in Fig. 4. 

Condition 1 automatically implies that the real and psendovalence eigenvalues agree for a chosen prototype configuration, as the eigenvalue determines the asymptotic decay of the orbitals. 

We have foimd that the region of existence of the stable periodic orbit is given by the condition e > x (/x, ), which can obviously be rewritten in the form > hkomil ) where the smooth function hhom behaves asymptotically as y/ fi /l2> The boundary of the region corresponds to the point Me on E, i.e. to a homoclinic loop of Oi. End of the proof. 

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