Periodic orbit stability

Proposition 1. For e sufficiently small the periodic orbit 7(t) of (7) survives as aperiodic orbit 7 (t) = rj(t)+0(e), of (10) having the same stability type as 7(t), and depending on e in a C2 manner. Moreover, the local stable and unstable manifolds Wlsoc( ye(t)) and Wfoc e(t)) of 7 (t) remain also e-close to the local stable and unstable manifolds WfoMt)) and Wfioc( (t)) of ft), respectively. 

The amplitude of the periodic orbits is therefore determined by the linear stability with respect to perturbations transverse to the orbit. In this sense, the leading term in expression (2.13), obtained by setting C = 0, treats the dynamics transverse to the orbit at the level of the harmonic approximation. The nonlinear stability properties appear thus as anharmonic corrections to the dynamics transverse to the orbit. These anharmonicities contribute to the trace formula by corrections given in terms of series in powers of the Planck constant involving the coefficients C , which can be obtained as Feynman diagrams [14, 31]. 

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... 

Moreover, the periodic orbits are here of neutral stability. 

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. 

The cases of hyperbolic-without-reflection and hyperbolic-with-reflection stability have to be distinguished. In both cases, the trajectories in the neighborhood of the periodic orbit trace out hyperbolic paths in the Poincare section, but if the stability is hyperbolic with reflection, the trajectories cross over between the branches of the hyperbola on each iteration. 

Periodic-orbit theory provides the unique semiclassical quantization scheme for nonseparable systems with a fully chaotic and fractal iepeller. As we mentioned in Section II, the different periodic orbits of the repeller have quantum amplitudes weighted by the stability eigenvalues, and the periodic-orbit amplitudes interfere among each other as described by the zeta function. The more unstable the periodic orbit is, the less it contributes in (2.24). Therefore, only the least unstable periodic orbits play a dominant role. 

The eigenfunctions associated with the resonances have been obtained via wavepacket propagation. They appear to be localized along the symmetric-stretch periodic orbit 0, with a number of nodes equal to n and even under the exchange of iodine nuclei. Due to the relative stability of the symmetric-stretch orbit, we have thus here a system where the hypothesis of the orbit 12 representing the RPO, that is, resonant periodic orbit, does not hold. 

Upon convergence, the eigenvalues of dF/dx (the characteristic or Floquet multipliers FMt) are independent of the particular point on the limit cycle (i.e. the particular Poincare section or anchor equation used). One of them, FMn, is constrained to be unity (Iooss and Joseph, 1980) and this may be used as a numerical check of the computed periodic trajectory the remaining FMs determine the stability of the periodic orbit, which is stable if and only if they lie in the unit circle in the complex plane ( FM, < 1,1 i = n - 1). The multiplier with the largest absolute value is usually called the principal FM (PFM). When (as a parameter varies) the PFM crosses the unit circle, the periodic orbit loses stability and a bifurcation occurs. 

Periodic orbits also explain the long-lived resonances in the photodissociation of CH.30N0(S i), for example, which we amply discussed in Chapter 7. But the existence of periodic orbits in such cases really does not come as a surprise because the potential barrier, independent of its height, stabilizes the periodic motion. If the adiabatic approximation is reasonably trustworthy the periodic orbits do not reveal any additional or new information. Finally, it is important to realize that, in general, the periodic orbits do not provide an assignment in the usual sense, i.e., labeling each peak in the spectrum by a set of quantum numbers. Because of the short lifetime of the excited complex, the stationary wavefunctions do not exhibit a distinct nodal structure as they do in truly indirect processes (see Figure 7.11 for examples). 

Poliak, E. (1981). Periodic orbits, adiabaticity and stability, Chem. Phys. 61, 305-316. 

Only the orbits infinitesimally close to the steady state may be considered stable, according to Liyapunov s theory of stability. However, at a finite distance from the steady state, two neighboring points belonging to two distinct cycles tend to be far apart from each other because of differences in the period. Such motions are called stable in the extended sense of orbital stability. The average concentrations of X and Y over an arbitrary cycle are equal to their steady-state values (Xs = 1 and Y.. A 1). Under these conditions, the average entropy production over one period remains equal to the steady-state entropy production. 

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. 

An unstable periodic orbit is one-dimensional, being of dimension two less than the energy surface in systems with two DOFs. In an n-DOF system the energy surface is of dimension 2n — 1. In such systems, Wiggins showed that the analog of unstable periodic orbits is the so-called normally hyperbolic invariant manifold (NHIM) of dimension 2n — 3 [20,21]. Trajectories slightly displaced from an NHIM can be analyzed using a many-dimensional stability analysis. The... 

In order to gain some experience with the new concepts introduced above, we will now discuss the stability properties of the fixed points and periodic orbits of the logistic mapping. The following is also a more in-depth presentation of the period doubling scenario briefly discussed in Section 1.2. 

Since 1 is a local attractor, to prove the theorem it remains only to show that it is a global attractor. This is taken care of by the Poincare-Bendixson theorem. As noted previously, stability conditions preclude a trajectory with positive initial conditions from having 0 or 2 in its omega limit set. The system is dissipative and the omega limit set is not empty. Thus, by the Poincare-Bendixson theorem, the omega limit set of any such trajectory must be an interior periodic orbit or a rest point. However, if there were a periodic orbit then it would have to have a rest point in its interior, and there are no such rest points. Hence every orbit with positive initial conditions must tend to j. (Actually, two-dimensional competitive systems cannot have periodic orbits.) Figure 5.1 shows the X1-X2 plane. 

Throughout the chapter, one assumes that the equilibria and periodic orbits that occur are hyperbolic. This means that local stability is determined from the linearization. Of course one knows this only after making the linearized computations. It is simply that nothing can be said in the... 

The question of which values of yield the periodic orbit is not resolved by the theorem. Generically, the derivative of A at 5 = 0 is not expected to be zero. In that case there would be an interval of values on one side of flf for which there would be a periodic solution in the positive cone see Figure 7.2. This affects the stability as well. A sample computation is given in the next section. 

The differential equations were solved for a variety of values of less than a. The program was run for considerable time and the last 100 points saved. If the limiting periodic orbit were asymptotically stable, these points would be near the periodic orbit - equal as well as the eye can determine. These periodic orbits, corresponding to different parameters and hence to different systems of differential equations, were then plotted on a single three-dimensional graph (Figure 8.1). This illustrates the stability. 

All of the comments raised in Chapter 3 in connection with a similar bifurcation apply. The computations needed to determine the direction of bifurcation (which side of ml) and the stability are formidable. However, for particular parameter values one can solve the differential equations numerically and exhibit the periodic orbit. Figure 3.1 shows the time course of a sample problem, and Figure 3.2 shows the projection of the periodic or the coexisting periodic orbits onto each of the possible pairs of variables. 

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