Attractive periodic orbit

If, in addition, the vector field is analytic, then it can be shown (see [ZS]) that there must be an attracting periodic orbit. 

Theorem C.3. A monotone dynamical system cannot have a nontrivial attracting periodic orbit. 

Proof of Theorem C.3. If there were an attracting periodic orbit, then one could find a point x in its domain of attraction such that x periodic orbit. As / is a limit point of the positive orbit through x, there exists T > 0 such that Xrest point by Theorem C.2(b), contradicting our assumption that it converges to a nontrivial periodic orbit. ... 

Theorem C.7 bears a strong resemblance to the Poincare-Bendixson theorem stated in Chapter 1. It will be used in Chapter 4 for the case where (C.l) is a competitive system, that is, for a system (C.l) where -/ is cooperative. Note that the omega (alpha) limit set of a competitive system is the alpha (omega) limit set of the time-reversed cooperative system, so Theorems C.5, C.6, and C.7 apply to competitive systems. Unlike cooperative systems, competitive systems can have attracting periodic orbits. For more on the Poincare-Bendixson theory of competitive and cooperative systems in see [S3], [SWl], and [ZS]. 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

A positive value of X arises because the trajectories in state space for chaotic behavior are diverging in the mean. Conversely, adjacent trajectories in a system which possesses a globally attracting limit cycle will converge. Values of X for periodic systems can be obtained by perturbing the reactor from a periodic state and observing the rate of convergence back to the periodic orbit. In Fig. 4 is shown the result of such an... 

The concepts of attraction, neutrality and repulsion can immediately be generalized to periodic orbits. Suppose that xq is the seed of a periodic orbit of length n, i.e. = xq. Then, clearly, xq is a fixed point... 

By a nontrivial periodic orbit we mean a periodic orbit that is not a rest point. Such an orbit is attracting if the omega limit set of each point of some neighborhood of the periodic orbit is the periodic orbit. 

The second important bifurcation that is connected with a stability change in a stationary state is the /fop/bifurcation. At a Hopf bifurcation, the real parts of two conjugate complex eigenvalues of J vanish, and as Hopf s theorem ensures, a periodic orbit or limit cycle is bom. A limit cycle is a closed loop in phase space toward which neighboring points (of the kinetic representation) are attracted or from which they are repelled. If all neighboring points are attracted to the limit cycle, it is stable otherwise it is unstable (see Ref. 57). The periodic orbit emerging from a Hopf bifurcation can be stable or unstable and the existence of a Hopf bifurcation cannot be deduced from the mere fact that a system exhibits oscillatory behavior. Still, in a system with a sufficient number of parameters, the presence or absence of a Hopf bifurcation is indicative of the presence or absence of stable oscillations. 

We perturb now the above two-body problem by adding to the model the gravitational attraction from a major planet (for example Jupiter), which we assume that revolves around the Sun in a circular orbit with constant angular velocity n. The study of the periodic orbits will be made in the rotating frame that rotates with constant angular velocity n. The new Hamiltonian has the form... 

At least in presence of periodic orbits on the 2-torus, drifting and pinning can coexist with positive measure see Figure 3.5(b). Indeed the pinning region of the attracting equilibrium (A) is now trapped between two unstable periodic orbits. The unstable periodic orbits also bound the basin of attraction of a stable periodic orbit. The stable periodic orbit signifies stable unbounded drift of the spiral tip across the lattice periodic perturbation pattern,... 

The idea that the vibrational enhancement of the rate is due to the attractive potential for excited vibrational states of the reactant is closely related to the observation made long ago based on transition state theory [25,26]. Poliak [25] found that for vibrationally highly excited reactants the repulsive pods (periodic orbit dividing snrface) is way out in the reactant valley, and the corresponding adiabatic barrier is shallow Based on this theory one can explain why dynamical thresholds are observed in reactions with vibrationally excited reactants. The simplicity of the theory and its snccess for mostly collinear reactions has a real appeal. However, to reconcile the existence of a vibrationally adiabatic barrier with the captnre-type behavior - which seems to be supported by the agreement of the calculated and experimental rate coefficients [23] -needs further study. 

There are many ways to display the solution to our general maps. One way is to plot successive values of x and y as dots on the screen. Many solutions will move toward a point and remain there. Others will settle into a periodic orbit or will move off toward infinity. The interesting cases are the chaotic ones that remain confined to a limited region but whose orbits produce a strange attractor with intricate fractal structure. You can choose the starting values of x and y arbitrarily within the basin of attraction, but you should discard the first few iterates because they probably lie off the attractor. 

Finally, consider the case where = 0. All of the available energy is trapped in the vibrational coordinate, so this case corresponds to maximum-amplitude vibrational motion within each isomer, with no motion along the reaction coordinate. Thus, there are two one-dimensional periodic orbits that sit at = q, the potential minima along the reaaion coordinate. All motion at this fixed energy must pass within the interior of one or both of these surfaces. Using the Pollak/Pechukas terminology, these orbits are attractive PODS 2 (the PODS nomenclature is explained more extensively later). 

The surface t v-1 is a particular example of what Wiggins has referred to as a hyperbolic manifold and what De Leon and Ling have termed a normally invariant hyperbolic manifoldd Hyperbolic manifolds are unstable and constitute the formal multidimensional generalization of unstable periodic orbits. Hyperbolic manifolds, like PODS, can be either repulsive or attractive. - If motion near a hyperbolic manifold falls away without recrossing it in configuration space, the hyperbolic manifold is said to be repulsive. On the other hand, it is often the case that motion near a hyperbolic manifold will cross it several times in configuration space as it falls away, and in this case it is said to be attractive. 

Therefore, if a structurally stable system has an attractive quasiminimal set — a strange attractor, then periodic orbits will be dense in it. 

The set of all points of the phase space whose trajectories converge to L as t —> +00 (—cx)) is called the stable (unstable) manifold of the periodic orbit. They are denoted by W and W , respectively. In the case where m = n, the attraction basin of L is Wf. In the saddle case, W is (m + l)-dimensional if m is the number of multipliers inside the unit circle, and is (p + 1)-dimensional where p is the number of multipliers outside of the unit circle, p = n — m — 1. In the three-dimensional Cctse, Wl and are homeomorphic either to two-dimensional cylinders if the multipliers are positive, or to the Mobius bands if the multipliers are negative, as illustrated in Fig. 7.5.1. In the general case, they are either multi-dimensional cylinders diffeomorphic to X S, or multi-dimensional Mobius manifolds. 

Fig. 10.2.4. Saddle-node periodic orbits in (a) the cycle L is stable in the interior region and unstable in the exterior region. When hp < 0, it is attractive for the point inside it, and repelling for outer trajectories (b).

If the first Lyapunov value L is positive, then for small fJL>0, the equilibrium state O is unstable and any other trajectory leaves a small neighborhood U of the origin. When fx <0, the equilibrium state is stable. Its attraction basin is bounded by an unstable periodic orbit of diameter /i which contracts... 

Let us examine next the bifurcations of the system (11.5.1) in the multidimensional case. If Li < 0 (Fig. 11.5.4), then when // < 0, the equilibrium state O is stable (rough focus when p < 0, and a weak focus aX p = 0) and it attracts all trajectories in a small neighborhood of the origin. When > 0 the point O becomes a saddle-focus with a two-dimensional unstable manifold and an m-dimensional stable manifold. The edge of the unstable manifold is the stable periodic orbit which now attracts all trajectories, except those in the stable manifold of O. One multiplier of the periodic orbit was calculated in Theorem 11.1, this is po p) = 1 — 47r /a (0) -h o p). To find the others we... 

If Li > 0, the phase portraits are depicted in Fig. 11.5.5. Here, when // < 0, there exists a stable equilibrium state O (a focus) and a saddle periodic orbit whose m-dimensional stable manifold is the boundary of the attraction basin of O. As /i increases, the cycle shrinks towards to O and collapses into it at /i = 0. The equilibrium state O becomes a saddle-focus as soon as p increases through zero. 

Fig 11.5.5. A subcritical Andronov-Hopf bifurcation, (a) An attraction basin of a stable focus is bounded by a stable manifold of a saddle periodic orbit, (b) The periodic orbit narrows to the stable focus at /x = 0, and the latter becomes a saddle-focus (1,2). 

Since the contraction in the local map can be made arbitrarily strong and the derivative of the global map is bounded, the superposition T = To oTi inherits the contraction of the local map for all small p as well. It then follows from the Banach principle of contracting mappings (Sec. 3,15) that the map T has a unique stable fixed point on So- As this is a map defined along the trajectories of the system, it follows that the system has a stable periodic orbit in V which attracts all trajectories in V. The period of this orbit is the sum of two times the dwelling time t of local transition from Sq to S and the flight time from Si to Sq. The latter is always finite for all small p. It now follows from (12.1.4) that the period of the stable periodic orbit increases asymptotically of order tt/x/a This completes the proof. 

Let us consider next the bifurcation of the saddle-node periodic orbit L in the case where the unstable manifold is a Klein bottle, as depicted in Fig. 12.3.1, i.e. when the essential map has degree m = -1. By virtue of Theorem 12.3, if is smooth, then a smooth invariant attracting Klein bottle persists when L disappears. In its intersection with a cross-section So, the flow on the Klein bottle defines a Poincare map of the form (see (12.2.26))... 

Starting with any (x, y), a trajectory of system (12.4.8) converges typically to an attractor of the fast system corresponding to the chosen value of x. This attractor may be a stable equilibrium, or a stable periodic orbit, or of a less trivial structure — we do not explore this last possibility here. When an equilibrium state or a periodic orbit of the fast system is structurally stable, it depends smoothly on x. Thus, we obtain smooth attractive invariant manifolds of system (12.4.8) equilibrium states of the fast system form curves Meq and the periodic orbits form two-dimensional cylinders Mpo, as shown in Fig. 12.4.6. Locally, near each structurally stable fast equilibrium point, or periodic orbit, such a manifold is a center manifold with respect to system (12.4.8). Since the center manifold exists in any nearby system (see Chap. 5), it follows that the smooth attractive invariant manifolds Meqfe) and Mpo( ) exist for all small e in the system (12.4.7) [48]. 

We assume throughout this section that the saddle values are negative in both saddles. In this case, no more than one periodic orbit can bifurcate from the heteroclinic cycle. Moreover, this unique orbit is stable (attracting). 

But alas most of what has been described so far concerning density theory applies in theory rather than in practice. The fact that the Thomas-Fermi method is capable of yielding a universal solution for all atoms in the periodic table is a potentially attractive feature but is generally not realized in practice. The attempts to implement the ideas originally due to Thomas and Fermi have not quite materialized. This has meant a return to the need to solve a number of equations separately for each individual atom as one does in the Hartree-Fock method and other ab initio methods using atomic orbitals. 

Polarizability increases from top to bottom in any column of the periodic table. As the principal quantum number ( ) increases, the valence orbitals become larger. This reduces the net attraction between valence electrons and the nucleus. 

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