Resonant fixed point

If the system has symmetries (as is the case with the restricted problem), usually the symmetric periodic orbits survive (but not always ). The resonant fixed points that survive correspond to monoparametric families of elliptic periodic orbits, in the rotating frame. These families bifurcate from the circular family, at the corresponding circular resonant orbits. From the above analysis we come to the conclusion that out of the infinite set of resonant elliptic periodic orbits of the two-body problem, with the same semimajor axes and the same eccentricities, but different orientations, as shown in Figure 15, only a finite number survive as periodic orbits in the rotating frame, and in most cases only two, usually, but not always, are symmetric. 

Fig. 10.5.2. A resonant fixed point with six separatrices. The angle between eau h pair is equal to 7t/3,...

We have seen in Sec. 10.4 that in the case of weak resonance cj = 2nM/N N > the stability of the critical fixed point is, in general, determined by the sign of the first non-zero Lyapunov value. The same situation applies to the critical case of an equilibrium state with a purely imaginary pair of characteristic exponents. However, there is an essential distinction, namely, for a resonant fixed point only a finite number which does not exceed N—3)/2 of the Lyapunov values is defined. The question of the structure of a small neighborhood of the fixed point in the case where all Lyapunov values vanish is difficult, so we do not study it here. Instead, we consider two examples. 

In fact, resonant fixed points are not restricted to only saddles and stable (completely unstable) points. An example of the other structure is given by the map... 

The method of exchange-luminescence [46, 47] is based on the phenomenon of energy transfer from the metastable levels of EEPs to the resonance levels of atoms and molecules of de-exciter. The EEP concentration in this case is evaluated by the intensity of de-exciter luminescence. This technique features sensitivity up to-10 particle/cm, but its application is limited by flow system having a high flow velocity, with which the counterdiffusion phenomenon may be neglected. Moreover, this technique permits EEP concentration to be estimated only at a fixed point of the setup, a factor that interferes much with the survey of heterogeneous processes associated with taking measurements of EEP spatial distribution. 

We now consider the phenomenon of entrainment (the development of resonances) on the torus (Meyer, 1983). When (and if) the off-diagonal band in Fig. 6 crosses the diagonal [Figs. 6(c) and 6(f)], there exist points whose images fall on themselves they are fixed points of the map we study. These points lie on periodic trajectories that are locked on the torus. Such trajectories appear in pairs in saddle-node bifurcations and are usually termed subharmonics . When this occurs there is no quasi-periodic attractor winding around the torus surface, but the basic structure of the torus persists the invariant circle is patched up from the unstable manifolds of the periodic saddle-points with the addition of the node-periodic point (Arnol d, 1973, 1982). As we continue changing some system parameter the periodic points may come to die in another saddle-node bifurcation (see Fig. 5). Periodic trajectories thus... 

E. This double -1 point is yet another codimension-two bifurcation, which will be discussed in detail later. Another period 1 Hopf curve extends from point F through points G and H. F is another double -1 point and, as one moves away from F along the Hopf curve, the angle at which the complex multipliers leave the unit circle decreases from it. The points G and H correspond to angles jt and ixr respectively and are hard resonances of the Hopf bifurcation because the Floquet multipliers leave the unit circle at third and fourth roots of unity, respectively. Points G and H are both important codimension-two bifurcation points and will be discussed in detail in the next section. The Hopf curves described above are for period 1 fixed points. Subharmonic solutions (fixed points of period greater than one) can also bifurcate to tori via Hopf bifurcations. Such a curve exists for period 2 and extends from point E to K, where it terminates on a period 2 saddle-node curve. The angle at which the complex Floquet multipliers leave the unit circle approaches zero at either point of the curve. 

The degree of freedom (q,p) has a resonant term Vcosq. There exist unstable fixed points ( = 7i,p = 0), and the separatrix orbits connecting them. The separatrix orbits of the nonlinear resonance are given by the following ... 

According to Chirikov [23J, the onset of chaos is associated with the overlap of neighboring nonlinear resonances. The overlap criterion, which bears the qualitative significance, uses the model of isolated resonances. Each resonance is characterized by its width, the maximum distance (in the action variable) from the elliptic fixed point The overlap means that the sum of the widths of two neighboring resonances is equal to the distance between two fixed points of these isolated resonances. We start with the pendulum Hamiltonian, which describes an isolated 1 N resonance under the periodic perturbation of frequency Q ... 

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... 

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. 

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). 

According to the Poincare-Birkhoff fixed point theorem all resonant tori break up for arbitrarily small perturbations. If the rotation number is p/q the perturbation leaves q pairs of hyperbolic and elliptic periodic orbits. The unstable hyperbolic orbits are embedded in a layer filled by aperiodic, chaotic orbits that do not stay on an invariant surface, but cover a finite non-zero volume of the phase space in a chaotic layer around the original resonant torus. The elliptic points, however, are wrapped around by new concentric tori that form islands of regular orbits within the chaotic band (Fig. 2.5). 

It is very important to stress that changes in the geometrical shape of the integration domain can induce bifurcations in the drift velocity field [31, 47, 50, 52]. Let us consider, for example, the drift velocity field computed for an elliptical domain with major axis o = 3A and minor axis b = a/1.1. As shown in Fig. 9.13(a), instead of the stable limit cycle of the resonance attractor in the circular domain of radius Rg, = 1.5A we have two pairs of fixed points where the drift velocity vanishes. In each pair, one fixed point is a saddle and the other one is a stable node. Depending on the initial conditions, the spiral wave approaches one of the two stable nodes. Trajectories of the spiral center obtained by numerical integration of the Oregonator model (9.1) are in perfect agreement with the predicted drift... 

The theoretically predicted destruction of the resonance attractor in response to deviations from the circular shape of the integration domain has been confirmed experimentally within the light-sensitive BZ medium. A spiral wave was exposed to uniform illumination proportional to the total gray level obtained in an elliptical integration domain. Fig. 9.13(b) shows the resonant drift mediated during global feedback control. The spiral wave drifts towards a stable node of the drift velocity field. Close to this fixed point the drift velocity becomes very slow. Thus, the experimentally observed termination of the spiral drift at certain positions in a uniform medium is explained in the framework of the developed theory of feedback-mediated resonant drift. 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

The normal mode representation of the phase space trajectories contains the same information as the local mode representation. However, the resonance region on the normal mode phase space map contains the local mode trajectories (la, lb 2a, 2b) and the stable fixed points Ca and C t,. The trajectories contained within the resonance zone are not free to explore the entire 0 < tp < n range whereas the trajectories outside the resonance zone do explore the 0 < ip < n range and are therefore classified as normal mode trajectories. The fixed point B (Iz = I = +2) is unstable, because it lies on a separatrix, and is located at the north pole of the normal mode polyad phase sphere. The stable fixed point A (7Z = — I = —2) is located at the south pole. 

Figure 9.16 shows the evolution of the local mode polyad phase spheres for H2O as I increases from 1 (N = vs + va = 1) to 3 (N = vs + va = 5). As I increases the lowest energy levels sequentially pass through the unstable fixed point B, depart from the normal mode resonance region, and become local mode states. At I = 1 (part (a)) there are only 2 levels and both are on the normal mode side of the separatrix. At I = 3 (part (e)) there are 6 levels and the lowest 4 of these have departed the resonance region and are local mode states. 

Figure 9.15 Evolution of the polyad phase sphere from the local mode to the normal mode limit as the strength of the 1 1 coupling term (antithetical to the local mode limit) is increased from 0 (part a) to oo (part f). As the coupling term, <5 in Eq. (9.4.174) increases from 0, first in part (c) one trajectory (level 4, at highest E) falls through the unstable fixed point into the normal mode region (antisymmetric stretch) eventually, in part (e), the resonance zone fills the entire phase sphere finally, in part (f), the normal mode limit is reached (from Xiao and Kellman, 1989).

The fact that these compounds were obtained as diastereoisomeric mixtures [the configuration of the (—)-menthyl unit is fixed, the configuration of the tin center may be either (R) or (5)], varying from 46 54 for 37 to 40 60 for 39 for the two diastereoisomers, indicates that the diastereose-lection during the reduction step is small. Evidence for the anticipated intramolecular Sn—N coordination in 37 and 38 was obtained from the H and C NMR spectra of these compounds. The increased (compared to the values of menthylmethylnaphthyltin hydride) J( C— Sn) and J( H— Sn) values of the menthyl and hydride resonances, respectively, points to an increased j-orbital participation in these bonds, while the decreased (again compared to the values of menthylmethylnaphthyltin hydride) J( C— Sn) and J( H— Sn) values for the methyl resonance points to an decreased j-orbital participation in this bond. These... 

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