Resonant invariant curves

All points on a resonant invariant curve (circle) are r-multiple fixed points the point comes to the initial position after r rotations along the angle 2 on the 2-torus. (A i = 2irs and A 2 = 2ttv). It can be easily seen that the unperturbed mapping (73) can be obtained from the generating function... 

A non-resonant invariant curve survives as closed invariant curve, due to the KAM theorem (see Figure 14b). 

Figure 14- (a) The resonant and non resonant invariant curves of the integrable... 

On a resonant invariant curve, out of the infinite set of r-multiple fixed points, only a finite (even) number survive, half stable and half unstable, as a consequence of the Poincare-Birkhoff fixed point theorem (Arnold and Avez, 1968 Lichtenberg and Lieberman, 1983), as shown schematically in Figure 14b. 

The resonant invariant curves correspond to the resonant elliptic periodic orbits, in the rotating frame. 

A resonant n/n = p/q elliptic periodic orbit is a multiple fixed point on the resonant invariant curve. The angle i) changes during one iteration by... 

We remark that all the fixed points on a resonant invariant curve of the Poincare map correspond to elliptic motion of the small body, with the same semimajor axis a, such that n/n is rational, and the same eccentricity e. They differ only in the orientation, which means that all these orbits have different values of w, as shown in Figure (15). 

The SSE phase-space portrait shown in Fig. 6.5 reminds us of the phase-space portraits of the kicked rotor presented in Chapter 5. In Fig. 6.5 we can identify resonances and sealing invariant curves. In Chapter 5 we saw that resonance overlap in the standard mapping defines a sudden percolation transition when for K > Kc the seahng invariant... 

According to the KAM theorem (Guckenheimer and Holmes, 1983), for sufficiently small e, the non resonant invariant circles survive the perturbation as nearly circular invariant curves. These invariant curves represent nearly elliptic orbits of the small body that are not periodic both in the rotating frame and the inertial frame. 

If the mapping (11.6.2) is the Poincare map of an autonomous system of differential equations, then the invariant curve corresponds to a two-dimensional smooth invariant torus (see Fig. 11.6.3). It is stable if L < 0, or it is saddle with a three-dimensional unstable manifold and an (m -h 2)-dimensional stable manifold if L > 0. Recall from Sec. 3.4, that the motion on the torus is determined by the Poincare rotation number if the rotation number v is irrational, then trajectories on the torus are quasiperiodic with a frequency rate u] otherwise, if the rotation number is rational, then there are resonant periodic orbits on a torus. 

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-... 

Note that both the saddles and nodes appearing inside the resonant wedge (called Arnold tongue, sometimes) lie on the invariant curve (stable if Li < 0 or unstable if L >0). Since the only stable invariant curve that can go through a saddle is its unstable manifold, and since the only unstable curve that can also go through a saddle is its stable manifold, it follows that inside the resonance zone the invariant curve is the union of the separatrices of saddles (imstable separatrices if Li < 0, or stable separatrices if L > 0) that terminate at the nodes. 

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,... 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

For this value of the perturbing parameter a lot of orbits are still invariant tori. Some resonant curves are displayed surrounding some elliptic points and a chaotic, though well confined, zone is generated by the existence of the hyperbolic point at the origin. 

The second curve corresponds to a weak chaotic orbit of initial conditions (0.4839,0) while the third one corresponds to a regular invariant torus of initial conditions (2.5,0) and the lowest one to a resonant curve of initial conditions (1.2,0). 

It was pointed out previously that the preceding analysis was inapplicable to media which contained heavy absorbers (So S,), especially if large, broad resonances appeared in the absorption cross-section curves. Unfortunately these are the materials of greatest interest to reactor tech-nology. Reactor cores invariably possess relatively high concentrations... 

In the spectra of CdSe, three modes at 258, 359 and 950 cm are clearly observed. In general, the LO and TO phonons are observed along with the surface modes in polar nanocrystals in resonance Raman spectra and/or surface enhanced Raman spectra [275]. However, LO and TO modes are observed simultaneously only in randomly oriented nanoparticles. Resonance Raman Spectra (RRS) of CdTe nanoparticles give band due to Longitudinal optical (LO) phonons at 170 cm (LO), 340 cm (2LO) and 510 cm (3LO) mode frequency is found to shift due to quantum confinement effect and confined phonons are observed using surface enhanced Raman spectroscopy [275]. Transverse optic (TO) phonon is reported at 145 cm and its position is invariant with decreasing particle size as the dispersion curve for TO phonon branch is almost fiat [275]. In CdSe nanoparticles, LO phonons are reported in the range 180- 200 cm, wheras in ZnSe at 140 cm [Ref. 275 and references therein]. Thus, the Raman spectra observed in the present work well identifies the phonons in these nanoparticles. 

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