Closed orbits existence

Now that we know how to rule out closed orbits, we turn to the opposite task finding methods to establish that closed orbits exist in particular systems. The following theorem is one of the few results in this direction. It is also one of the key theoretical results in nonlinear dynamics, because it implies that chaos can t occur in the phase plane, as discussed brief y at the end of this section. 

It s time to change gears. So far in this chapter, we have focused on a qualitative question Given a particular two-dimensional system, does it have any periodic solutions Now we ask a quantitative question Given that a closed orbit exists, what can we say about its shape and period In general, such problems can t be solved exactly, but we can still obtain useful approximations if some parameter is large or small. 

A collision with a Mars-sized object may have resulted in the formation of the Earth s moon. Our moon is by no means the largest satellite in the solar system, but it is unusual in that it and the moon of Pluto are the largest moons relative the mass of the planets they orbit. Geochemical studies of returned lunar samples have shown that close similarities exist between the bulk composition of the moon and the Earth s mantle. In particular, the abimdances of sidero-... 

Proof. First suppose is a complex torus T. Taking a basis of V, we may suppose we are in the above situation. (If some coordinates are 0, we replace P by a subspace.) We may also assume that Y is the unique closed orbit in the closure of x. The uniqueness follows from the existence of T -invariant polynomial which separates two disjoint T -invariant closed subsets (Theorem 3.3). 

Fig. 10.10 proves that a close connection exists between the classical mechanics and the quantum mechanics of the simple one-dimensional two-electron model. On the basis of the evidence provided by Fig. 10.10, there is no doubt that classical periodic orbits determine the structure of the level density in an essential way. The key element for establishing the one-to-one correspondence between the peaks in R and the actions of periodic orbits is the scaling relations (10.3.10). Similar relations hold for the real helium atom. Therefore, it should be possible to establish the same correspondence for the three-dimensional helium atom. First steps in this direction were taken by Ezra et al. (1991) and Richter (1991). 

The main difference between the (3" structure compared to the / phase is the direction of the strong intermolecular interactions. Due to the smaller anion size the interaction directions are at 0°, 30°, and 60°, respectively, instead of face-to-face (90°) overlaps [335]. The more complicated interstack interaction results in a more anisotropic band structure with ID and 2D energy bands. There exists considerable disagreement between different band-structure calculations which might be caused by small differences in the transfer integral values [332, 335, 336]. One calculated FS based on the room temperature lattice parameters is shown in Fig. 4.27a [335]. Small 2D pockets occur around X and two ID open sheets run perpendicular to the a direction. In contrast, the calculation of [332] (not shown) revealed a rather large closed orbit around the F point. 

According to Theorem C.6, the limit set can be deformed to a compact invariant set A, without rest points, of a planar vector field. By the Poin-care-Bendixson theorem, A must contain at least one periodic orbit and possibly entire orbits which have as their alpha and omega limits sets distinct periodic orbits belonging to A. Using the fact that A is chain-recurrent, Hirsch [Hil] shows that these latter orbits cannot exist. Since A is connected it must consist entirely of periodic orbits that is, it must be an annulus foliated by closed orbits. Monotonicity is used to show... 

Even for systems that have nothing to do with mechanics, it is occasionally possible to construct an energy-like function that decreases along trajectories. Such a function is called a Liapunov function. If a Liapunov function exists, then closed orbits are forbidden, by the same reasoning as in Example 7.2.2. 

Dulac s Criterion Let x = f(x) be a continuously differentiable vector field defined on a simply connected subset R of the plane. If there exists a continuously differentiable, real-valued function g(x) such that V (gx) has one sign throughout R, then there are no closed orbits lying entirely in R. 

When /z=0, there s a stable limit cycle at r=l, as discussed in Example 7.1.1. Show that a closed orbit still exists for p > 0, as long as p is sufficiently small. 

The estimates used in Example 7.3.1 are conservative. In fact, the closed orbit can exist even if p. > 1. Figure 7.3,3 shows a computer-generated phase portrait of (1) for /z = 1. In Exercise you re asked to explore what happens for larger /z, and in particular, whether there s a critical p beyond which the closed orbit disappears. It s also possible to obtain some analytical insight about the closed orbit for small p (Exercise 7.3.9). 

Can we conclude that there is a closed orbit inside the trapping region No There is a fixed point in the region (at the intersection of the nullclines), and so the conditions of the Poincard-Bendixson theorem are not satisfied. But if this fi xed point is a repeller, then we can prove the existence of a closed orbit by considering... 

Assume the hypotheses of Dulac s criterion, except now suppose that R is topologically equivalent to an annulus, i.e., it has exactly one hole in it. Using Green s theorem, show that there exists at most one closed orbit in R. (This result can be useful sometimes as a way of proving that a closed orbit is unique.)... 

When (7) holds, the Poincare-Bendixson theorem implies the existence of a closed orbit somewhere in the punctured box. ... 

The argument above proves the existence of a closed orbit, and almost proves its uniqueness. But we haven t excluded the possibility that Ply) = y on some in-... 

By process of elimination, we conclude that all fixed points must be sinks or saddles, and closed orbits (if they exist) must be stable or saddle-like. For the case of fixed points, we now verify these general conclusions explicitly. 

Photoelectron and microwave spectroscopy have been used to detect and optimize the formation of various phosphaalkynes, RC=P, in gas-phase flow-pyrolysis procedures. Photoelectron spectra show that the ionization processes observed relate to electron-removal from orbitals of essentially n(CP) character. A close analogy exists between the ligand properties of such phosphaalkynes towards transition metals and those of conventional alkynes. ... 

Assume also that x (a) is a critical point of Eq. (29). The eigenvalues Pi( ) Piia),..., pjv(a) will now also depend on the parameter a. If for some values of a, say a < Gq, the critical point is stable, and if in addition a pair of complex conjugate eigenvalues pi(a), p2(o) cross the imaginary axes transversely (d Repi(a)/da a=ao5 0) then we say that a Hopf bifurcation takes place at the value a = Gq. If a Hopf bifurcation occurs in Eq. (29) then there exists a one-parameter family of periodic solutions for a in the neighborhood of ao with a period near 27r/ Im pi(ao)l- K the flow attracts to the critical point x (ao) when a = Gq, then x°(ao) is called a vague attractor. For this case the family of closed orbits is contained m a >Go and the orbits are of attracting type. ... 

As it will be seen in next Section, metal cluster structures are closely related to the number of electrons and orbitals existing in the cluster core. In general. 

The pure Volterra-Lotka model leads to a mathematically marginal case The singular point (= stationary state) is a center encircled by closed orbits. The latter are non-stationary solutions depending on the initial conditions in analogy to the solutions of undamped heuniltonian systems in theoretical mechanics. Therefore the original model is sensitive to small perturbations which may have different causes [4.1, 23-26, 30, 31]. On the other hand, Volterra-Lotka cycles have been observed by biologists in nature for several entirely different species [4.21, 22, 27-29]. It has been stressed that the existence of such cycles could depend on the availabihty of appropriate facilities for migration [4.28, 31]. 

Then, in the extended phase space the direct product of the phase space and the parameter space) near the origin there exists a uniquely defined -smooth invariant surface of the form p = 0(a ), V (0) = 0, such that each its intersection with the plane p = constant consists of a set of closed orbits of the system (11.5.17), lying in a neighborhood of the origin at the given p. 

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