Invariant circle

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

As we further change the parameter R, the hysteresis interval ends (the invariant circle stops existing) and the only attractor is the stable periodic frequency locked solution N. Both sides of the unstable manifold of the sad e-type frequency locked solution are attracted to N (Point G, inset 2e). 

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

One fixed point, stable or unstable, in the stroboscopic map corresponds to a periodic solution of period t. If there are two distinct points, each mapping into the other, we have a periodic solution of twice the period. Figure 30, drawn with a continuous fill-in, illustrates the way in which different kinds of periodic solution can coexist. The invariant circle (actually ovoid with a pointed... 

Beyond its applicability in the simplification of the computations, the stroboscopic representation greatly simplifies the recognition of patterns in the transient response of periodically forced systems. A sustained oscillation appears as a finite number of repeated points, while a quasi-periodic response appears as an invariant circle (see Figs. 3, 4, 6 and 9). 

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... 

There exists a quantity (the rotation number) which is independent of the particular stroboscopic section we take and characterizes the entire torus. For a radial parameterization of the invariant circle the rotation number is defined—and may be computed—as the limit... 

The first interesting phenomenon we observe is the breaking of the smooth invariant circle picture when the node-periodic points on the circle become foci [Figs. 8(a) and 8(b)]. The basic structure of the trajectories remains the same, but the object we are now studying is no longer a well-defined circle ... 

In our computation of invariant circles of maps the main cost lies in the... 

An alternative formulation for the torus-computing algorithm is to solve for an invariant circle along with a nonlinear change of coordinates that makes the action of the stroboscopic map conjugate to a rigid rotation on the circle. This is equivalent to the parameterization... 

It is based on Denjoy s theorem, and ris the rotation number. This algorithm, implemented by Chan (1983) computes invariant circles with irrational rotation numbers. We may, of course, discretize and solve for the whole invariant surface and not just for a section of it. Instead of having to integrate the system equations, we will then be solving for a much larger number of unknowns resulting from the additional dimension we had suppressed in the shooting approach we used. 

Aronson, D. G., Chory, M. A., Hall, G. R. and McGehee, R. P., 1982, Bifurcations from an invariant circle for two-parameter families of maps of the plane a computer assisted study. Comm. Math. Phys. 83, 303-354. 

Iooss, G., Ameodo, A., Coullet, P. and Tresser, G, 1980, Simple computations of bifurcating invariant circles for mappings. Lect. Notes Math. 898,192-211. 

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. 

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. 

Numerical computation of invariant circles of maps (with I.G. Kevrekidis, L.D. Schmidt, and S. Pelikan). Physica 16D, 243-251 (1985). 

One more important ingredient in discussing the transport process in mixed phase space is so-called cantori. They are invariant sets in which the motion has an irrational frequency. They resemble invariant circles, but they have an infinite number of gaps in them. The existence was proposed by Percival [30] and Aubry [31]. They gave an explicit example, and a proof of their existence has been given afterwards [32-34]. 

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. 

For the moment, let us not identify the points corresponding to R = 0 when the values of

phase space of system (10.5,14) becomes a semi-cylinder. On the invariant circle R — 0 there are six equilibrium states... 

Hence, by a small perturbation of the original map (11.7.14) we can transform (in some new variables) the map of an invariant circle into a rotation with a constant angle. If we change a little bit the value u in (11.7.17) so that it becomes a rational number u = M/N then the map on the circle R = 1 assumes the form... 

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,... 

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