Two-dimensional invariant torus

A straightforward generalization of two-dimensional bifurcations was developed soon after. So were some natural modifications such as, for instance, the bifurcation of a two-dimensional invariant torus from a periodic orbit. Also it became evident that the bifurcation of a homoclinic loop in high-dimensional space does not always lead to the birth of only a periodic orbit. A question which remained open for a long time was could there be other codimension-one bifurcations of periodic orbits Only one new bifurcation has so far been discovered recently in connection with the so-called blue-sky catastrophe as found in [152]. All these high-dimensional bifurcations are presented in detail in Part II of this book. 

Same as Case 3 but Li(0) > 0, The stable periodic orbit becomes unstable when an unstable two-dimensional invariant torus shrinks into it. 

Although the phase space trajectories appear as simple curves on the two-dimensional Iz,ip phase space diagram (the 0 coordinate is suppressed) most trajectories are actually quasiperiodic. The actual trajectories he on the 2-dimensional surface of a 3-dimensional invariant torus in 4-dimensional phase space. Fig. 9.14 shows such a torus. Any point on the surface of the torus is specified by two angles, 0 and. The 0 and circuits about the torus are shown, respectively, as large and small diameter circles. The diameter of the 0... 

The KAM theorem, it should be noted, has nothing to say about what happens when the strength of the perturbation increases. However, a considerable amount of experience has accumulated from detailed numerical calculations performed for many systems. One can visualise the results by studying Poincare sections if a cut is made across an invariant torus (see fig. 10.3) and a numerical calculation of trajectories is performed over a sufficiently long time, the stable orbits fill the deformed tori densely, and so result in closed curves in the two-dimensional cut, whereas the irregular or chaotic orbits yield a random speckle. 

Figure 95 shows topological portraits of the integrable case of normal series of a four-dimensional rigid body. This invariant is more complicated than those enumerated above. It should be noted that nonorientable edges of graphs F(Q) are absent (i.e., there are no asterisks). The two-dimensional surfaces P(Q) are homeomorphic either to a sphere or to a torus. 

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. 

We have already established in the last section that when the first Lyapunov value does not vanish, the passage over the stability boundary 9Jl p e) = 0 is accompanied by the appearance of an invariant two-dimensional torus (in the associated Poincare map this corresponds to the appearance of an invariant closed curve). If we are not interested in the behavior of the trajectories on the torus, we can restrict our consideration to the study of one-parameter families transverse to 9Jl, In this case Theorem 11.4 in Sec. 11.6, gives a complete description of the bifurcation structure. In order to examine the... 

The closed invariant curve Wq for the Poincare map on the cross-section is the loci of intersection of an invariant two-dimensional torus W with the cross-section. The torus is smooth if the invariant curve is smooth, and it is non-smooth otherwise. If the original non-autonomous system does not have a global cross-section, then other configurations of W are also possible, as... 

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