Quasiperiodic orbit

It is readily seen that the set of equations (76) consists of three equations of motion in the real variables ReIm c, w. If, (x) = constant, chaos in the system does not appear since the set (76) becomes a two-dimensional autonomous system. The maximal Lyapunov exponents for the systems (75) and (72)-(74) plotted versus the pulse duration T are presented in Fig. 36. We note that within the classical system (75) by fluently varying the length of the pulse T, we turn order into chaos and chaos into order. For 0 < T < 0.84 and 1.08 < 7) < 7.5, the maximal Lyapunov exponents Li are negative or equal to zero and, consequently, lead to limit cycles and quasiperiodic orbits. In the points where L] = 0, the system switches its periodicity. The situation changes dramatically if,... 

Marston, C.C. and Wyatt, R.E. (1985). Semiclassical theory of resonances in 3D chemical reactions ILResonant quasiperiodic orbits for F and H2, J. Chem. Phys. 83, 3390. 

Aperiodic long-term behavior means that there are trajectories which do not settle down to fixed points, periodic orbits, or quasiperiodic orbits as t —> . For practical reasons, we should require that such trajectories are not too rare. For instance, we could insist that there be an open set of initial conditions leading to aperiodic trajectories, or perhaps that such trajectories should occur with nonzero probability, given a random initial condition. 

Although in a weakly time-dependent flow all resonant tori disappear together with some of the nearly resonant tori around them, the Kolmogorov-Arnold-Moser theorem ensures that infinitely many invariant surfaces survive a small perturbation. For sufficiently small e the remaining invariant surfaces formed by quasiperiodic orbits, so called KAM tori, still occupy a non-zero volume of the phase space. The condition for a torus to survive a given perturbation is that its rotation number should be sufficiently far from any rational number so that the inequality... 

Thus some of the fluid elements move on aperiodic chaotic trajectories and others on quasiperiodic orbits. The quasiperiodic orbits are invariant surfaces in the phase space that form the boundaries of the chaotic layers and limit the motion of the chaotic trajectories. There is a similar structure around each elliptic periodic orbit resulting from broken resonant tori that are also surrounded by invariant tori forming isolated islands inside the chaotic region. 

Thus, the time-dependence of the flow generates chaotic trajectories that will enhance the mixing of fluid within these regions. However, the KAM tori formed by the remaining quasiperiodic orbits separate the domain into a set of disconnected regions with no advec-tive transport between them. Therefore, when the time-dependence is weak the fluid is only mixed within narrow layers around the resonant streamlines of the original time-independent flow. The areas... 

Fig. 10.12. Some examples of quasiperiodic orbits outside the plane perpendicular to the magnetic field (after P.F. O Mahony et al. [558]).

Figure 12 (Top) two trapped quasiperiodic orbits at fixed energy superimposed on potential energy contours for the De Leon-Berne Hamiltonian at = 0.65 (see Figure 23 for the potential energy surface). (Below) The Poincare map for this system at this energy, with the surface of section defined at fixed <]2 with p2 > 0. The trajectories above are connected with their corresponding map locations below. Note that whereas these relatively regular motions lie on tori, most of the Poincare map is broken up, indicating that most motions at this energy are chaotic. Reprinted with permission from Ref. 119.

Semiclassical Theory of Resonances in 3D Reactions. II. Resonant Quasiperiodic Orbits for F + H2. 

Given the quasiperiodic orbit embedded in the continuum one can proceed to identify in configuration space the coordinates (u,v,w). In principle then, using classical mechanics one is able to identify also in a coplanar reaction a best set of adiabatic coordinates. It may also happen that the bend and stretch frequencies are similar in magnitude. In such a case the periodic reduction method cannot work. However usually the rotational motion will still be much slower than the stretches and bends and so could still be treated adiabatically. Such a scheme may be termed a quasiperiodic reduction method.It has been applied successfully for adiabatic barriers and wells.However there is a price to pay, it is more tedious. 

A different example is the H3 system. Here the ground state of H3 is an equilateral triangle. Recent experiments of Carrington and Kennedy indicate that has a high density of quasibound states embedded in the continuum above the H ground electronic state dissociation limit. These states or at least some of them should be representable as stable periodic and quasiperiodic orbits. 

As for attractive minimal sets, it follows from Pugh s theorem that they are structurally unstable. Although the minimal sets composed of recurrent and limit-quasiperiodic orbits are by far not key players in the nonlinear dynamics, quasiperiodic motions have always been of major interest because they model many oscillating phenomena having a discrete spectrum. 

---