Resonant torus

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). 

Figure 2.5 Sketch of the typical structures associated with the breakup of a resonant torus on the Poincare section in a weakly time-periodic flow producing a set of new elliptic and hyperbolic points.

Fig 11.7.2. An example of a stable resonant torus with five pairs of periodic orbits on it. The torus becomes non-resonant when the periodic orbits coalesce in pairs and disappear. 

In such a case we have the so-called soft loss of stability. The newly established regime inside the attracting spot may be either a new equilibrium state, a periodic trajectory, a non-resonant torus, or even a strange attractor (a situation generally referred as instant chaos). The latter option is possible when Oeo bas three zero eigenvalues (see [18], or [129] for systems with symmetry). 

For the elucidation of chemical reaction mechanisms, in-situ NMR spectroscopy is an established technique. For investigations at high pressure either sample tubes from sapphire [3] or metallic reactors [4] permitting high pressures and elevated temperatures are used. The latter represent autoclaves, typically machined from copper-beryllium or titanium-aluminum alloys. An earlier version thereof employs separate torus-shaped coils that are imbedded into these reactors permitting in-situ probing of the reactions within their interior. However, in this case certain drawbacks of this concept limit the filling factor of such NMR probes consequently, their sensitivity is relatively low, and so is their resolution. As a superior alternative, the metallic reactor itself may function as the resonator of the NMR probe, in which case no additional coils are required. In this way gas/liquid reactions or reactions within supercritical fluids can be studied... 

In the previous subsection, the forcing frequency was exactly twice the natural oscillatory frequency. Thus the motion around one oscillation gives exactly two circuits of the forcing cycle for one revolution of the natural limit cycle. The full oscillation of the forced system has the same period as the autonomous cycle and twice the forcing period. The concentrations 0p and 6r return to exactly the same point at the top of the cycle, and subsequent oscillatory cycles follow the same close path across the toroidal surface. This is known as phase locking or resonance. We can expect such locking, with a closed loop on the torus, whenever the ratio of the natural and forcing... 

In between the resonance horns are regions of the parameter plane for which the response is quasi-periodic. Note that it is even possible for the frequencies to have a simple ratio and yet for the system to lie outside the corresponding resonance horn if the amplitude is raised. Figure 13.15 shows two time series for forcing with oj/oj0 = 10/1. At low forcing amplitude, rr = 0.005, we have phase locking and a simple if rather crumpled limit cycle. With rf = 0.01, however, the response is quasi-periodic a few cycles are shown and demonstrate quite well how the trajectory begins to wind around the torus. 

If the CFM is formed by a complex pair, the unit circle can be crossed at some point (actually at two points simultaneously) off the real axis. Generally this will correspond to the bifurcation from a stable limit cycle to a quasi-periodic motion on a torus. In special cases, however, where the crossing point corresponds to the fcth complex root of - 1, the limit cycle bifurcates to a phase-locked cycle (closed loop on the torus) corresponding to period-fc resonance such as that described ip 13.3. 

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... 

It is interesting to consider the shapes of the subharmonic trajectories that lock on the torus in the various entrainment regions of order p/q. The subharmonic period 4 at the 4/3 resonance horn is, for example, a three-peaked oscillation in time [Fig. 7(a)] and has three closed loops in its phase-plane projection [Fig. 7(b)], while the subharmonic period 4 at the 4/ 1 resonance is a single-peaked, single-loop oscillation [Figs. 7(d) and 7(e)]. A subharmonic period 2 at the 2/3 resonance is also included in Figs. 7(g) and 7(h). Multipeaked oscillations observed in chemical systems (Scheintuch and Schmitz, 1977 Flytzani-Stephanopoulos et al., 1980) may thus result from the interaction of frequencies of local oscillators. Such trajectories are the nonlin-... 

Small-order resonance horns (p, q small) and particularly those with 1 p, <7 4 are comparatively wide and easier to locate computationally through algorithms that will locate the periodic entrained trajectories. These algorithms, however, will be inadequate for a complete analysis of our systems since (at least as FA — 0) periodic trajectories appear in disconnected isolas. The motivation behind the construction of our torus-computing algorithm is to provide a means of study of this two-parameter bifurcation diagram that can continue smoothly both within the resonance horns and in the region of quasi-periodicity that separates or—from another point of view—unites them. 

The most important characteristic in our test cases, however, is that within the 1/1 and the 2/1 resonance horns the torus will break as FA increases. In all models this happens when the unstable source period 1 that existed within the torus hits the saddle-periodic trajectories that lie on the torus. This occurs through a saddle-node bifurcation in the 1/1 resonance horn [Fig. 8(d)], and through an unstable period doubling in the 2/1 resonance [Fig. 8(c)]. After these bifurcations the basic structure of the torus has collapsed, and we are left only with the stable entrained periodic trajectories. 

Let us examine more closely what occurs on the right-hand boundary of the 1/1 resonance horn [Fig. 9(a)]. In a sequence of one-parameter bifurcation diagrams with respect to oj/co0, each taken at a successively higher forcing amplitude FA, we observe that as FA increases, the bifurcation point to a torus changes. The point of exit of the Floquet multipliers of the periodic... 

At interesting phenomenon occurs in the case of other resonance horns we have studied it for the case of the 3/1 resonance. The torus pattern breaks when the subharmonic periodic trajectories locked on it for small FA decollate from the torus as FA increases. We are left then with two attractors a stable period 3 and a stable quasi-periodic trajectory. This is a spectacular case of multistability (co-existence of periodic and quasi-periodic oscillations). The initial conditions will determine the attractor to which the system will eventually converge. This decollation of the subharmonics from the torus was predicted by Greenspan and Holmes (1984). They also predicted chaotic trajectories close to the parameter values where the subharmonic decollation occurs. 

An example is shown in figure 3 for section AA near the bottom of the 2/1 resonance horn of figure 2. As the frequency is increased from left to right, the torus becomes phase locked as a pair of period 2 saddle nodes develop on it. The saddle nodes then separate with the saddles alternating with the node and the invariant circle is now composed of the unstable manifolds of the saddles whereas the stable manifolds of the saddles come from the unstable period 1 focus in the middle of the circle and from infinity. As the frequency is increased further, the saddles rotate around the circle and recombine with their neighbouring nodes in another saddle-node bifurcation. 

The three standard local codimensional-one bifurcations are the saddle-node, Hopf, and period doubling bifurcations and several have been continued numerically for this model and appear in figure 2. We have chosen not to show the curves of focus-node transitions because they do not represent any changes in stability, only changes in the approach to the steady behaviour. The saddle-node bifurcations that occur during phase locking of the torus at low amplitudes continue upward and either close upon themselves as in the case of the period 3 resonance horns or the terminate in some codimension-two bifurcation. 

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. 

FIGURE 8 (a) Detail of the tip of the 3/1 resonance horn illustrating typical way in which period 3 resonance horns close around a point with Floquet multipliers at the third root of unity (point F). (fa) and (c) The saddle-node pairings change from section AA to section BB and the unstable manifolds of the period 3 saddles no longer make up a phase locked torus. (d) A one-parameter vertical cut through the third root of unity point. The three saddles coalesce with the period one focus that is undergoing the Hopf bifurcation. 

A feature that, to our knowledge when we discovered it, had not been seen before in forced oscillators (Marek and his co-workers have also observed it (M. Marek, personal communication)) is the folding that occurs in the left side of the 3/2 and 2/1 resonance horns. Within these folds there are two sets of stable nodes and two sets of saddles, so that bistability between the two sets exists. There are also cases of bistability between subharmonic responses of period 3 and a torus in the top of the period 3 resonance horns. In addition to the implication of bistability, the fold in the side of the 3/2 resonance horn may be of mathematical significance. Aronson et al. (1986) put forth the mathematical conjecture that if the period 3 resonance horn is a simple disc-... 

The corresponding trajectories can be best visualized as motion restricted to a two-dimensional torus, as shown in Fig. 1. If the frequency ratio, or the winding number ( i/( 2, is a rational number, the two DOFs are in resonance and an individual trajectory will close on itself on the torus. By contrast, if coi/a)2 is an irrational number, then as time evolves a single trajectory will eventually cover the torus. The motion in the latter case is called conditionally periodic. 

Random phase approximation D = (K2 + b2)/2 works well for strong coupling (e.g., b > 10) and strong nonlinearity (e.g., K = 4.0), that waits for higher-order correction by Fourier path method [11,15,16], With K > 1.0, diffusion coefficients approach some constants as b —> 0 due to the breakup of the last KAM torus of each standard map, while with K < 0.9 and smaller b, they are expected to be evaluated by the stochastic pump or three resonance model and their extensions [12,17,18]. 

One can Immediately ask, what If there Is not such an EBK torus for the coupled system The empirical result Is that the method may well work anyway Table 111 shows results (26) obtained by adiabatic quantization of the Hase ( ) HCC two-degrees-of-free-dom problem. Away from the 5 2 resonances (see Fig. 5) the adiabatic and Hase results are In accord, but the adiabatic method also successfully quantizes the resonance zones. Another Illustration Is given In Table IV where a two-degrees-of-freedom model of HOD Is adlabatlcally quantized above the classical dissociation threshold... 

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