Bifurcation of resonant periodic orbit

This study [14] has shown that a period-doubling bifurcation associated with the Fermi resonance occurs in this subsystem at the energy E = 3061.3 cm-1 (with Egp = 0). Below the Fermi bifurcation, there exist edge periodic orbits of normal type, which are labeled by ( i, 22 -)normai- At the Fermi bifurcation, a new periodic orbit of type (2,1°, ->Fermi appears by period doubling around a period of 2T = 100 fs. This orbit is surrounded by an elliptic island that forms a region of local modes in phase space. Therefore, another family of edge periodic orbits of local type are bom after the Fermi bifurcation that may be labeled by the integers (n n , -)iocai- They are distinct... 

Figure 19. Scattering resonances of 2F collinear models of the dissociation of H3 obtained by Manz et al. [143] on the Karplus-Porter surface and by Sadeghi and Skodje on a DMBE surface [132], The energy is defined with respect to the saddle point. The dashed lines mark the bifurcations of the transition to chaos 1, 5, and 3 are the numbers of shortest periodic orbits, or PODS, in each region.

If the system has symmetries (as is the case with the restricted problem), usually the symmetric periodic orbits survive (but not always ). The resonant fixed points that survive correspond to monoparametric families of elliptic periodic orbits, in the rotating frame. These families bifurcate from the circular family, at the corresponding circular resonant orbits. From the above analysis we come to the conclusion that out of the infinite set of resonant elliptic periodic orbits of the two-body problem, with the same semimajor axes and the same eccentricities, but different orientations, as shown in Figure 15, only a finite number survive as periodic orbits in the rotating frame, and in most cases only two, usually, but not always, are symmetric. 

In fact, no common upper bound exists on the number of the periodic orbits which can be generated from a fixed point of a smooth map through the given bifurcation. If the smoothness r of the map is finite, the absence of this upper estimate is obvious because it follows from the proof of the last theorem that to estimate the number of the periodic orbits within the resonant zone 1/ = M/N the map must be brought to the normal form containing terms up to order (AT — 1). In this case the smoothness of the map must not be less than (iV — 1). Hence, we can estimate only a finite number of resonant zones if the smoothness is finite. 

Ezra G S 1996 Periodic orbit analysis of molecular vibrational spectra-spectral patterns and dynamical bifurcations in Fermi resonant systems J. Chem. Phys. 104 26... 

To answer Prof. Marcus s question, we may therefore conclude that the natural motions of the system are the short-time periodic orbits. Those that arise from the symmetric-stretch bifurcations depend on the frequency ratio local modes in the 1 1 case, 7-shaped orbits at the 3 2 instability, horseshoes at the 2 1 resonances, and so on. 

See, for example, the following and references contained therein E. L. Sibert 111, W. P. Reinhardt, and J. T. Hynes, /. Chem. Phys., 81, 1115 (1984). Intramolecular Vibrational Relaxation and Spectra of CH and CD Overtones in Benzene and Perdeuterobenzene. S. P. Neshyba and N. De Leon,. Chem. Phys., 86, 6295 (1987). Qassical Resonances, Fermi Resonances, and Canonical Transformations for Three Nonlinearly Coupled Oscillators. S. P. Neshyba and N. De Leon,. Chem. Phys., 91, 7772 (1989). Projection Operator Formalism for the Characterization of Molecular Eigenstates Application to a 3 4 R nant System. G. S. Ezra, ]. Chem. Phys., 104, 26 (1996). Periodic Orbit Analysis of Molecular Vibrational Spectra Spectral Patterns and Dynamical Bifurcations in Fermi Resonant Systems. Also see Ref. 6. 

As /i increases within a resonant zone other periodic orbits with the same rotation number M/N may appear. In some cases, the boundary of the resonant zone can lose its smoothness at some points, like in the example shown in Fig. 11.7.4 here, the resonant zone consists of the union of two regions D and Z>2 corresponding to the existence of, respectively, one and two pairs of periodic orbits on the torus. The points C and C2 in Fig. 11.7.4 correspond to a cusp-bifurcation. At the point S corresponding to the existence of a pair of saddle-node periodic orbits the boundary of the resonant zone is non-smooth. 

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. 

The boundary of the resonant zone corresponds to a coalescence of the stable and unstable periodic orbits on the invariant circle, i.e. to the saddle-node bifurcation of the same type we consider here. Besides, if there were more than two periodic orbits, saddle-node bifurcations may happen at the values of parameters inside the resonant zone. By the structure of the Poincare map (12.2.26) on the invariant curve,... 

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