Poincare resonances

MSN. 142. T. Petrosky and I. Prigogine, Poincare resonances and the limits of quantum mechanics, Phys. Lett. A 182, 5-15 (1993). 

MSN. 153.1. Prigogine and T. Petrosky, Poincare Resonances and the Extension of Classical and Quantum Mechanics, in Nonlinear, deformed and irreversible quantum systems, H. D. Doebner, V. K. Dobrev, and P. Nattermann, eds.. World Scientific, Singapore, pp. 3-21, 1995. 

Next we construct At by their action on the product a a. A first idea would be to keep the distributive relation A a a = (Ataj)(Atai) = a A analogous to Eq. (10). But the product a Ai contains terms nonanalytic at A, = 0 [see discussion below Eq. (26)] due to Poincare resonances. 

MSN. 146. I. Prigogine and T. Petrosky, Poincare s resonances and extension of classical and quantum mechanics, in Proceedings, 12th Symposium Energy Engineering Sciences, Argonne National Laboratory, 1994, pp. 8-16. 

Because there is no resonance between the particle and the field, C/t expandable in a power series in X, around A, = 0. This means the system is integrable in the sense of Poincare. 

As a result, we have Poincare s resonance singularity at = i for i > 0 in the series expansion of in X. The Friedrichs model discussed above may become nonintegrable. 

Figure 3. Poincare surface of section of the T-shaped HeBr2 at the Br-Br vibrational state V = 15, showing a 4 1 resonance. [From A. A. Granovsky et al., J. Chem. Phys. 108, 6282 (1998).]...

Figure 4. Poincare surface of section of molecular isomerization of cyclobutanone (C4H6O) for the total reaction energy E = O.Ola.u, showing a 3 1 resonance.

Figure 10. Poincare surface of section for collinear OCS relaxation at = 20,000 cm . It shows three major quasi-periodic regions, the resonance islands, the location of the dividing surface for intramolecular energy transfer, and a typical turnstile. [From M. J. Davis, J. Chem. Phys. 83, 1016 (1985).]...

Figure 17. The Poincare surface of section for T-shaped Hel2 with the initial vibrational state of I2 given by v = 10. Two bottlenecks to intramolecular energy transfer are shown, together with a 5 1 resonance zone and the dissociation dividing surface. From top to bottom the figures show how trajectories escape the first and then the second intramolecular botdenecks. The bottom panel shows trajectories passing the separatrix for dissociation. [From M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986).]...

The numerical results give us the confidence to attempt an analytical calculation of critical ionization fields. This can be done with some success by computing the widths of the resonances apparent in Fig. 7.5 and using the widths as input to Chirikov s overlap criterion as discussed in Section 5.2. The analytical method allows us to compute critical ionization fields for many initial conditions no and field parameters and u without the need to inspect a sequence of Poincare sections in each particular case. Presently, however, the available analytical methods are not very accurate. For rough estimates of classical critical ionization fields, however, the currently available analytical techniques are very useful. 

If the resonant tori, which are the invariant tori whose rotational numbers are rational, are broken under perturbations, the pairs of elliptic and hyperbolic cycles are created in the resonance zone. This fact is known as a result of the Poincare-Birkhoff theorem [4], which holds only if the twist condition, Eq. (2), is satisfied. Around elliptic cycles thus created, new types of tori, which are... 

As we know from theorem 3, due to Poincare, in the general case of a non-degenerate unperturbed Hamiltonian Ho(p) the construction of the normal form can not be completely performed in a consistent manner. However, we are still allowed to perfom a suitable construction in domains where some resonances may be excluded, provided we perform a Fourier cutoff on the perturbation. We shall discuss this method in connection with Nekhoroshev s theorem. 

On a resonant invariant curve, out of the infinite set of r-multiple fixed points, only a finite (even) number survive, half stable and half unstable, as a consequence of the Poincare-Birkhoff fixed point theorem (Arnold and Avez, 1968 Lichtenberg and Lieberman, 1983), as shown schematically in Figure 14b. 

We remark that all the fixed points on a resonant invariant curve of the Poincare map correspond to elliptic motion of the small body, with the same semimajor axis a, such that n/n is rational, and the same eccentricity e. They differ only in the orientation, which means that all these orbits have different values of w, as shown in Figure (15). 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

Figure 2.5 Sketch of the typical structures associated with the breakup of a resonant torus on the Poincare section in a weakly time-periodic flow producing a set of new elliptic and hyperbolic points.

Fig. 10.16. Poincare surface of section of a strongly driven Morse oscillator at a laser field strength of 0.1 atomic units, with the frequency tuned to a two-photon resonance condition with excitation from the ground state to the fourth excited level. The plot indicates the presence of a resonance structure, away from which the motion lies on regular tori (after J.-P. Connerade et al. [584]).

Figure 10 Poincare map for two noniinearly coupled oscillators at fixed energy. The energetic periphery is labeled X. (A) The elliptic fixed point of an orbit with a),/o)2 = /i. (B, C) Elliptic fixed points of two distinct orbits with mjuiz - Vj. (D) Sep-aratrix surrounding the Wj/wj = Vi resonance zone. The separatrix pinches off at four hyperbolic fixed points, three of which are more or less clearly visible. (E) One of four elliptic fixed points of an orbit with V (the other three are not visi-...

In Figure 10 the chaotic region is extremely small. However, in Figures 11 and 12 we show a second system s phase space map as a function of en-ergy.35,119 xhjs system exhibits a mode-mode resonance at low energies, with a hyperbolic fixed point located near the center of the Poincare map. Note in Figure 11 that as the energy increases, the measure of quasiperiodic phase space decreases and approaches a limit in which most of the tori are destroyed, with... 

Once we have the kinetic equation, it is easy to show that we have irreversible processes and entropy production. It seems to me therefore very natural to consider that chemistry is indeed a very important example of nonintegrable Poincare systems, where the non-integrability is due to resonances. I hope that this new aspect will continue to be explored by future generations of physicists and chemists. ... 

If the mapping (11.6.2) is the Poincare map of an autonomous system of differential equations, then the invariant curve corresponds to a two-dimensional smooth invariant torus (see Fig. 11.6.3). It is stable if L < 0, or it is saddle with a three-dimensional unstable manifold and an (m -h 2)-dimensional stable manifold if L > 0. Recall from Sec. 3.4, that the motion on the torus is determined by the Poincare rotation number if the rotation number v is irrational, then trajectories on the torus are quasiperiodic with a frequency rate u] otherwise, if the rotation number is rational, then there are resonant periodic orbits on a torus. 

The invariant manifold depends continuously on p. At p = 0, it coincides with W, When /x < 0, it is the imion of the mist able manifold of the saddle periodic orbit L p) with the stable periodic orbit L p) (where L p) are the periodic orbits into which the saddle-node bifurcates ). In the case of torus, for p> 0, the Poincare rotation number on Tfj, tends to zero as /x -> +0. Thus, on the /x-axis there are infinitely many (practically indistinguishable as p -hO) resonant zones which correspond to periodic orbits on 7 with rational rotation numbers, as well as an infinite set (typically, a Cantor set) of irrational values of p for which the motion on is quasiperiodic. 

Therefore, in the orientable case, the Poincare rotation number on the torus depends monotonically on (see Sec. 4.4). Typically, each rational rotation number corresponds to an interval of values of fi (a resonant zone). In the simplest case, there exist only two periodic orbits on the torus in the... 

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