Lyapunov exponents periodic orbit

It is readily seen that the set of equations (76) consists of three equations of motion in the real variables ReIm c, w. If, (x) = constant, chaos in the system does not appear since the set (76) becomes a two-dimensional autonomous system. The maximal Lyapunov exponents for the systems (75) and (72)-(74) plotted versus the pulse duration T are presented in Fig. 36. We note that within the classical system (75) by fluently varying the length of the pulse T, we turn order into chaos and chaos into order. For 0 < T < 0.84 and 1.08 < 7) < 7.5, the maximal Lyapunov exponents Li are negative or equal to zero and, consequently, lead to limit cycles and quasiperiodic orbits. In the points where L] = 0, the system switches its periodicity. The situation changes dramatically if,... 

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.

In reaction dynamics, NHIMs with saddles would lose normal hyperbolicity as the energy of the vibrational modes increases at saddles. This is shown schematically in Fig. 32. Here, a saddle X of the potential function is displayed with its NHIM above in the phase space. When the reaction takes place with only a small amount of the energy in the vibrational modes, orbits go over the saddle where the vibrational motions are quasi-periodic. In Fig. 32, this is shown by the dotted arrow with tori on the NHIM. As the energy of the vibrational modes increases, however, orbits go over the saddle where the vibrational motions are chaotic because of the coupling among the vibrational modes. In Fig. 32, this is shown by the solid arrow with chaos (shown by the wavy line) on the NHIM. If the Lyapunov exponents of these chaotic motions become larger than those of the normal directions, the condition of normal hyperbolicity breaks down. 

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. 

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.

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. 

In this situation, a periodic variation of coolant flow rate into the reactor jacket, depending on the values of the amplitude and frequency, may drive to reactor to chaotic dynamics. With PI control, and taking into account that the reaction is carried out without excess of inert (see [1]), it will be shown that it the existence of a homoclinic Shilnikov orbit is possible. This orbit appears as a result of saturation of the control valve, and is responsible for the chaotic dynamics. The chaotic d3mamics is investigated by means of the eigenvalues of the linearized system, bifurcation diagram, divergence of nearby trajectories, Fourier power spectra, and Lyapunov s exponents. 

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