Lyapunov periodic orbit

The NHIM has a special structure due to the conservation of the center actions, it is filled, or foliated, by invariant d — l)-dimensionaI tori. More precisely, for d = 3 DoFs, each value of Jz implicitly defines a value of h by the energy equation Kcnf(0,/2,/3) = E. For three DoFs, the NHIM is thus foliated by a one-parameter family of invariant 2-tori. The end points of the parameterization interval correspond to Jz = 0 (implying qz = Pi = 0) and /s = 0 (implying q3=pz = 0), respectively. At the end points, the 2-tori thus degenerate to periodic orbits, the so-called Lyapunov periodic orbits. 

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 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-... 

Figure 7. Lyapunov exponents of the periodic orbits (n4, n ) = (5,6), (6,7) in the bending subsystem of Hamiltonian (3.14). The numerical error is -5 x 10-4 fs-1. (From Ref. 114.)...

At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. 

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 13. Scattering resonances of a two-degree-of-freedom collinear model of the dissociation of Hgl2 [10], The filled dots are obtained by wavepacket propagation, the crosses by equilibrium point quantization with (2.8), the dotted circles from periodic-orbit quantization with (4.16). The solid lines are the curves corresponding to the Lyapunov exponents, Im E = -(A/2)Xp(Re ), of the fundamental periodic orbits p = 0, 1,2. The dashed line is the spectral gap, Im = ftP(l/2 Ref). The long-short dashed line is the curve corresponding to the escape rate, Im E = (h/2)P( Re ).

Figure 16. Scattering resonances of the full rotational-vibrational Hamiltonian describing the dissociation of CO2 on a LEPS surface obtained by equilibrium point quantization with (2.8). The resonances with 7 = 0,..., 10 are given by dots. Their close vicinity explains the formation of hyphens , i.e., unresolved sequences of dots. Note that rotation is very slightly destabilizing in the present model. The successive hyphens are the bending progressions with V2 = 0,. .. 5. The solid line is given by the Lyapunov exponent of the symmetric-stretch periodic orbit 0 expressed as an imaginary energy.

The fact that there are exactly N Lyapunov exponents is easily understood in the case of a periodic orbit with period p. Periodicity means that the orbit returns to its starting point after p applications of M. In other words, Xp = xq. If we denote by U the product of the Jacobians... 

But for a chaotic mapping periodic orbits are dense. In other words, arbitrarily close to any nonperiodic orbit we find a periodic orbit. Thus, we can apply the above considerations. Therefore, in general, there are exactly N Lyapunov exponents that can be used to characterize the local rate of exponential divergence, and thus chaos itself. 

The collection of orbits and their properties listed in Table 10.1 turn out to be useful in Section 10.4.3, where we extract periodic orbit information firom the exact quantum spectrum of the one-dimensional model. The properties of the periodic orbits corroborate the claim that the onedimensional helium atom is completely chaotic All periodic orbits found so far are unstable with a positive Lyapunov exponent. 

The paper is organized as follows in Section 2 and 3 we define the Fast Lyapunov Indicator and give some examples on the 2 dimensional standard map and on a Hamiltonian model. The special case of periodic orbits will be detailed in 4 and thanks to a model of linear elliptic rotation we will be able to recover the structure of the phase space in the vicinity of a noble torus. The use of the FLI for detecting the transition between the stable Nekhoroshev regime to the diffusive Chirikov s one will be recalled in Section 5. In 6 and 7 we will make use of the FLI results for the detection of the Arnold s diffusion. 

Lega, E. and Froeschle, C. (2001). On the relationship between fast Lyapunov indicator and periodic orbits for symplectic mappings. Celest. Mech. and Dynamical Astronomy, 15 1-19. 

Fig. 6.18. The phase portraits of the synchronous attractor before the blowup (a) at 7) = 0.0282 and after (b) at 77 = 0.0287. Those parts of the attractors, which possesses negative iocal Lyapunov exponents are marked by the biack points. Period-3 unstabie periodic orbit is shown by broken red line.

As noted earlier, a necessary and sufficient condition for TST to be exact at a dividing surface is that any classical trajectory crossing the surface will never recross it. It is of practical interest to determine when TST must fail. Usually, the complete potential energy surface is not available so that one would want to develop local criteria for the failure of TST. Here we will shoxv that the stability properties of periodic orbits can be used to answer this question. Loosely speaking, a periodic orbit is defined as stable in the sense of Lyapunov if every trajectory originating at t=0 close enough to the periodic orbit remains close to the orbit for all time t. Obviously, TST cannot be exact if the pods is stable, since there are, by definition, trajectories crossing the pods that stay in its vicinity forever. For these trajectories and so TST is not exact. 

There is a regular closed stable trajectory near a shear rate 7 = 0.8 which must have no positive Lyapunov exponents. For this window of periodic orbits the period varies smoothly with 7 and the results are summarised in the Table(l). The periodic orbit is similar to that for 7 = 1.0 shown in Fig. (5) but with many less loops associated with wagging s. 

C Complex complicated motion of the alignment tensor. This includes periodic orbits composed of sequences of KT and KW motion with multiple periodicity as well as aperiodic, erratic orbits. The largest Lyapunov exponent for the latter orbits is positive, i.e., these orbits are chaotic. 

Quasiperiodic trajectories are a special case of Poisson-stable trajectories. The latter plays one of the leading roles in the theory of dynamical systems as they form a large class of center motions in the sense of Birkhoff (Sec. 7.2). Birkhoff had partitioned the Poisson-stable trajectories into a number of subclasses. This classification is schematically presented in Sec. 7.3. Having chosen this scheme as his base, as early as in the thirties, Andronov had undertaken an attempt to collect and correlate all known types of dynamical motions with those observable from physical experiments. Since his arguments were based on the notion of stability in the sense of Lyapunov for an individual trajectory, Andronov had soon come to the conclusion that all possible Lyapunov-stable trajectories are exhausted by equilibrium states, periodic orbits and almost-periodic trajectories (these are quasiperiodic and limit-quasiperiodic motions in the finite-dimensional case). 

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

If all Lyapunov values are equal to zero and the system is analytic, then the center manifold is also analytic, and all points on it, except O, are periodic of period two. This means that for the system of differential equations there exists a non-orientable center manifold which is a Mobius band with the cycle L as its median and which is filled in by the periodic orbits of periods close to the double period of L (see Fig. 10.3.2). 

Fig. 10.3.2. The center manifold of the primary periodic orbit L of an analytic system is a Mobius strip filled densely by periodic orbits of double period when all Lyapunov coefficients vanish.

Theorem 11.1. If the first Lyapunov value Li in (11.5.3) is negative, then for small /i < 0, the equilibrium state O is stable and all trajectories in some neighborhood U of the origin tend to O. When fx > 0, the equilibrium state becomes unstable and a stable periodic orbit of diameter y/Ji emerges see Fig. 11.5.1) such that all trajectories from U, excepting O, tend to it. 

If the first Lyapunov value L is positive, then for small fJL>0, the equilibrium state O is unstable and any other trajectory leaves a small neighborhood U of the origin. When fx <0, the equilibrium state is stable. Its attraction basin is bounded by an unstable periodic orbit of diameter /i which contracts... 

Note that since the map is defined for i > 0, we need only to look for non-negative roots R = 0 corresponds to an equilibrium state, and the positive roots correspond to periodic orbits of the system (11.5.13). Since we have already examined an equation of this type in the preceding section (Eq. (11.4.12)), when analyzed the period-two orbits emerging from a perioddoubling bifurcation in the case of zero Lyapunov value, we can simply reformulate the main results. 

Proof. Let us suppose, for definiteness, that the first Lyapunov value Li is negative. Then, the invariant curve exists when /x > 0. The resonance zone adjoining at the point /jL = = a o) corresponds to periodic orbits of period-... 

Observe that if the iV-th iteration of the circle map is an identity, then all points on the circle are structurally unstable with a multiplier equal to one. Moreover, all Lyapunov values of each point are equal to zero. This is an infinitely degenerate case. We saw in Sec. 11.3 that to investigate the bifurcations of structurally unstable periodic orbits with A — 1 first zero Lyapunov values it is necessary to consider at least -parameter families. It is now clear that to study bifurcations in this case one has to introduce infinitely many pa rameters. Moreover, it is seen from the proof of Theorem 11.5 that such maps can be obtained by applying a small perturbation to an arbitrary circle map... 

Remark. This statement remains valid (with obvious modifications) also in the case of the on-edge homoclinic loop to a degenerate saddle-node. In this case, /i is a vector of parameters (of dimension equal to the number of zero Lyapunov values plus one), and an additional bifurcation parameter e is introduced as before. A stable periodic orbit exists when the saddle-node disappears (the region /j> Dq m our notations), or when e > hhomi/j) fjL Do. Here, the surface e = hhomifJ ) corresponds to the homoclinic loop of the border saddle equilibrium Oi, as illustrated in Fig. 12.1.7. 

A periodic orbit collapses into a weak focus (the length of the periodic orbit shrinks to zero while it approaches the bifurcation point). This condition coincides with the condition defining the boundary of an equilibrium state with a single pair of pure imaginary eigenvalues provided that the Lyapunov value Li e) < 0. 

Note that the first Lyapunov value is always negative for a flip-bifurcation of any periodic orbit in the logistic map. Indeed, the Schwarzian derivative ... 

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