Stable periodic orbit

It is sometimes very usefiil to look at a trajectory such as the synnnetric or antisynnnetric stretch of figure Al.2.5 and figure A1.2.6 not in the physical spatial coordinates (r. . r y), but in the phase space of Hamiltonian mechanics [16, 29], which in addition to the coordinates (r. . r ) also has as additional coordinates the set of conjugate momenta. . pj. ). In phase space, a one-diniensional trajectory such as the aiitisymmetric stretch again appears as a one-diniensional curve, but now the curve closes on itself Such a trajectory is referred to in nonlinear dynamics as a periodic orbit [29]. One says that the aihiamionic nonnal modes of Moser and Weinstein are stable periodic orbits. 

Further insights into reaction dynamics can be obtained by analyzing the stability of classical trajectories. Presumably, stable periodic orbits will be restricted to KAM tori and therefore be nonreactive and unstable periodic orbits will provide information about the location of resonances and therefore some quahtative features of the intramolecular energy flow. 

Let us now consider the dynamics of the coupled system with Hamiltonian H = Hp, + Hp. Ju and Jp remain good quantum numbers for this Hamiltonian and are quantized according to Eq. (31). It is known that the dynamics of the coupled system is governed by the shape of its stable periodic orbits (POs) in the subspace of the normal coordinates involved in the Fermi... 

Proof. Let 7 = (a (/), (/), 0) be the orbitally asymptotically stable periodic orbit of period T given by Theorem 5.4. (We have already noted that if there are several orbits then one must be asymptotically stable, by our assumption of hyperbolicity.) Let the Floquet multipliers of 7, viewed as a solution of (3.1), be 1 and p, where 0periodic orbit, define p( 3) by... 

ZS] H.-R. Zhu and H. Smith (1994), Stable periodic orbits for a class of three dimensional competitive systems, Journal of Differential Equations 110 143-56. 

The studies of Wiesenfeld [28] and Lai et al. [43] on the classical dynamics of a one-electron atom in a sinusoidal external field provide a physically realistic example in which the presence of KAM tori surrounding stable periodic orbits leads to deviations from the generic behaviour characteristic of a hyperbolic scattering system as discussed in Sect. 2. Although this system (10) seems simple, further studies illuminating the mathematical structures behind the scattering process, e.g. calculation of the Liapunov exponents of the unstable trapped orbits and the fractal dimension of the trapped set, have yet to be performed. 

The Newtonian gravitational force is the dominant force in the N-Body systems in the universe, as for example in a planetary system, a planet with its satellites, or a multiple stellar system. The long term evolution of the system depends on the topology of its phase space and on the existence of ordered or chaotic regions. The topology of the phase space is determined by the position and the stability character of the periodic orbits of the system (the fixed points of the Poincare map on a surface of section). Islands of stable motion exist around the stable periodic orbits, chaotic motion appears at unstable periodic orbits. This makes clear the importance of the periodic orbits in the study of the dynamics of such systems. 

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,... 

A general necessary requirement for the applicability of Wigner-Dyson statistics to physical systems is that there should be no constants of motion other than the energy itself, to eliminate crossing of levels. In classical mechanics such systems are known as non-integrable or chaotic systems (no stable periodic orbits in phase space). Besides impurity scattering, scattering at boundaries can make the system... 

M. upper and lower bounds of stable periodic orbits... 

A totally diflFerent situation becomes possible in the case where the system does not have a global cross-section, and when is not a manifold. In this case (Sec. 12.4), the disappearance of the saddle-node periodic orbit may, under some additional conditions, give birth to another (unique and stable) periodic orbit. When this periodic orbit approaches the stability boundary, both its length and period increases to infinity. This phenomenon is called a hlue-sky catastrophe. Since no physical model is presently known for which this bifurcation occurs, we illustrate it by a number of natural examples. 

The bifurcations of periodic orbits from a homoclinic loop of a multidimensional saddle equilibrium state are considered in Sec. 13.4. First, the conditions for the birth of a stable periodic orbit are found. These conditions stipulate that the unstable manifold of the equilibrium state must be one-dimensional and the saddle value must be negative. In fact, the precise theorem (Theorem 13.6) is a direct generalization of the Andronov-Leontovich theorem to the multi-dimensional case. We emphasize again that in comparison with the original proof due to Shilnikov [130], our proof here requires only the -smoothness of the vector field. 

Fig. 8.1.4. The semi-stable periodic orbit L is the cj-limit of the separatrix of the outer saddle Oi, and the a-limit of the separatrix of the inner saddle O2.

Fig. 11.3.7. Scenario of a saddle-node bifurcation of periodic orbits in The stable periodic orbit and the saddle periodic orbit in (a) coalesce at /i = 0 in (b) into a saddle-node periodic orbit, and then vanishes in (c). The unstable manifold of the saddle-node orbit in (c) is homeomorphic to a semi-cylinder. A trajectory following the path in (b) slows down along a transverse direction near the virtual saddle-node periodic orbit (i.e., the ghost of the saddle-node orbit in (b)) so that its local segment is similar to a compressed spring.

Theorem 11.1. If the first Lyapunov value Li in (11.5.3) is negative, then for small /i < 0, the equilibrium state O is stable and all trajectories in some neighborhood U of the origin tend to O. When fx > 0, the equilibrium state becomes unstable and a stable periodic orbit of diameter y/Ji emerges see Fig. 11.5.1) such that all trajectories from U, excepting O, tend to it. 

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... 

Fig 11.5.4. supercritical Andronov-Hopf bifurcation in R . The stable focus (the leading manifold jg two-dimensional) in (a) becomes a saddle-focus in (b). A stable periodic orbit is the edge of the unstable manifold W. ... 

Inside the resonant wedge all trajectories on the invariant torus tend to a stable periodic orbit, which means that the dominating regimes here is a periodic one. Outside the wedge either a quasiperiodic regime or a periodic one of a very long period is established on the torus. Both are practically indistinguishable. Therefore, a transition over the boundary of a resonant... 

Theorem 12.1. The disappearance of the saddle-node equilibrium with the homoclinic loop results in the appearance of a stable periodic orbit of period... 

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. 

This situation is completely analogous to that we have in the study of a homoclinic loop to a saddle, which is considered in detail in Sec. 13.4. There (Lemma 13.4.1) we prove that the contracting map of the kind under consideration has a stable fixed point if and only if the single point of the image of the boundary of the domain of definition lies inside the domain. Thus, our system has a stable periodic orbit if and only if the point Me lies in the region... 

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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