Periodic orbits bifurcating

Founargiotakis, M., Farantos, S.C., Contopoulos, G., and Polymilis, C. (1989). Periodic orbits, bifurcations, and quantum mechanical eigenfunctions and spectra, J. Chem. Phys. 91, 1389-1401. 

System (3.3) has a unique equilibrium point E = (S, x ), since f is strictly increasing. A periodic orbit bifurcates from for t large. 

A new family of periodic orbits bifurcates from the T-periodic orbit x(t) of the original family, starting with the same period. Note that stability changes (critical points with respect to the stability). 

Since < 0, the saddle value is positive, and hence the periodic orbit bifurcating from the separatrix loop is unstable here. 

Fig. 13.7.2. The bifurcations of the homoclinic-8 in the symmetric case. An outward breakdown of both homoclinic loops gives birth to a large symmetric periodic orbit. When the loops split inwards, a periodic orbit bifurcates from each of the loops.

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... 

Figure C3.6.5 The first two periodic orbits in the main subhannonic sequence are shown projected onto the (c, C2) plane. This sequence arises from a Hopf bifurcation of the stable fixed point for the parameters given in the text. The arrows indicate the direction of motion, (a) The limit cycle or period-1 orbit at k 2 = 0.11. (b) The first subhannonic or period-2 orbit at k 2 = 0.095.

Figure C3.6.6 The figure shows tire coordinate, for < 0, of tire family of trajectories intersecting tire Poincare surface at cq = 8.5 as a function of bifurcation parameter k 2- As tire ordinate k 2 decreases, tire first subhannonic cascade is visible between k 2 0.1, tire value of tire first subhannonic bifurcation to k 2 0.083, tire subhannonic limit of tire first cascade. Periodic orbits tliat arise by tire tangent bifurcation mechanism associated witli type-I intennittency (see tire text for references) can also be seen for values of k 2 smaller tlian tliis subhannonic limit. The left side of tire figure ends at k 2 = 0.072, tire value corresponding to tire chaotic attractor shown in figure C3.6.1(a). Otlier regions of chaos can also be seen.

Even though the bifurcation behavior exhibits a Z-shaped curve, it is more complicated due to the existence of the HB. For example, upon ignition, the system is expected to oscillate because no locally stable stationary solutions are found (an oscillatory ignition). Time-dependent simulations confirm the existence of self-sustained oscillations [7, 12]. The envelope of the oscillations (amplitude of H2 mole fraction) is shown in circles (a so-called continuation in periodic orbits). 

Fig. 26.1a). At first, multistage ignitions and extinctions occur followed by a relaxation (long period) mode [7]. Oscillations die a few degrees below the ignition temperature at a saddle-loop infinite-period homoclinic orbit bifurcation point. This is an example where both ignition and extinction are oscillatory. 

Intramolecular dynamics and chemical reactions have been studied for a long time in terms of classical models. However, many of the early studies were restricted by the complexities resulting from classical chaos, Tlie application of the new dynamical systems theory to classical models of reactions has very recently revealed the existence of general bifurcation scenarios at the origin of chaos. Moreover, it can be shown that the infinite number of classical periodic orbits characteristic of chaos are topological combinations of a finite number of fundamental periodic orbits as determined by a symbolic dynamics. These properties appear to be very general and characteristic of typical classical reaction dynamics. 

For periodic orbits that undergo a bifurcation, some Lyapunov exponents may vanish so that the orbit becomes of neutral linear stability in the critical directions [32]. In such cases, the dynamics transverse to the periodic orbit... 

Due to the nonlinearities of the classical Hamiltonian, the periodic orbits undergo bifurcations at critical energies. At these bifurcations, the stability of the orbit changes and extra periodic orbits are created or existing ones annihilated [19]. These bifurcations have dramatic effects on the semiclassical amplitudes of the periodic orbits [49]. In particular, the comparison between the amplitudes of neutrally stable and unstable periodic orbits shows that the amplitude is expected to be globally lowered after a destabilization. 

Recent results show that even after the annihilation of periodic orbits at a bifurcation there remain traces of the annihilated periodic orbits in the quantum amplitudes [50]. These so-called ghost periodic orbits have been interpreted in terms of the complex periodic orbits created when the corresponding real periodic orbits are annihilated. Such phenomena have been experimentally observed for Rydberg atoms in a magnetic field [11,51]. We shall show below that such phenomena are also important in intramolecular and dissociation dynamics. 

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... 

Moreover, the Poincare mappings of (3.14) at values of Pc) fixed by the existence of (5, 6) and I4 + L5 = 0 show the presence of classically chaotic motions with a bifurcation at E - 6900 cm-1 (see Fig. 6). At this bifurcation, the periodic orbit (5, 6) becomes unstable because one of its Lyapunov exponents turns positive, as shown in Fig. 7. The periodic orbit (6, 7) destabilizes by a similar scenario around E - 7200 cm-1. These results show that the interaction between the bending modes leads to classically chaotic behaviors that destabilize successively the periodic orbits. For the bulk peri-... 

The theory of bifurcations shows that the different types of bifurcations can be described in terms of normal forms, which represent local expansions of the dynamics around the bifurcating periodic orbit [19, 32, 49]. The purpose of the above mapping is to describe the successive bifurcations of the symmetric-stretch periodic orbit, starting from low energies above the saddle point. Appropriate truncation of the Taylor series of the potential v(q) around <7 = 0, which corresponds to the location of the symmetric-stretch orbit, provides us with the normal forms of the bifurcations [144], The bifurcations relevant for the dissociation dynamics under discussion can be described by truncating at the sixth order in q,... 

Figure 9. Supercritical antipitchfork bifurcation scenario for symmetric XYX molecules on the left, bifurcation diagram in the plane of energy versus position in the center, typical phase portraits in some Poincare section in the different regimes on the right, the fundamental periodic orbits in position space.

At still higher energies, the elliptic island undergoes a typical cascade of bifurcations in which subsidiary elliptic islands of periods 6, 5, 4, 3 are successively created, which leads to the global destruction of the main elliptic island to the benefit of the surrounding chaotic zone. The cascade ends with a period-doubling bifurcation at Ed, above which the periodic orbit 0 is hyperbolic with reflection, and the main elliptic island has disappeared... 

The above scenario is accounted for by the normal form (4.9) truncated at fourth order in q with k = v = a = p = 0 and x < 0, taking p as the bifurcation parameter, which increases with energy (p thus plays a similar role as the total energy in the actual Hamiltonian dynamics). The antipitchfork bifurcation occurs at pa = 0. The fixed points of the mapping (4.8) are given by p = 0 and dv/dq = 0. Since the potential is quartic, there are either one or three fixed points that correspond to the shortest periodic orbits 0, 1, and 2 of the flow. 

As the energy increases in the interval E < E < Ea, the orbits Y and 2 progressively shift toward the symmetric-stretch orbit 0 and merge at the subcritical antipitchfork bifurcation. Just below this bifurcation, 1 and 2 are elliptic while 0 is still hyperbolic (without reflection). Between and Ea, the periodic orbits 1 and 2 may either remain of elliptic type or become hyperbolic in the energy interval [ , "1 such that E < Edd> < Edd < Ea. 

Differences between the lifetimes obtained from equilibrium point quantization and periodic-orbit quantization appear as the bifurcation is approached. The lifetimes are underestimated by equilibrium point quantization but overestimated by periodic-orbit quantization. The reason for the upward deviation in the case of periodic-orbit quantization is that the Lyapunov exponent vanishes as the bifurcation is approached. The quantum eigenfunctions, however, are not characterized by the local linearized dynamics but extend over larger distances that are of more unstable character. 

Figure 14. Period-energy diagram of the fundamental periodic orbits 0, I, and 2 in Hgl2 with the bifurcations of the transition region marked by dashed lines.

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