Arnold tori

The oscillating orbits of the second type. They also belong to Arnold tori of quasi-periodic motions, but with a major difference with respect to the bounded motions they have an infinite number of very close approaches, as close as desired, of the two bodies of the binary and, even if the radius-vectors remains forever bounded, the velocities are unbounded. 

For given values h and c, we have thus an eight-dimensional phase space of the eight Delaunay s parameters, and the motion remains in the seven-dimensional subspace given by the condition H = h. In that subspace we find infinitely many four-dimensional Arnold tori and a four-dimensional manifold of collisions. 

Many Arnold tori intersect the collision manifold along a onedimensional submanifold and have thus a two-dimensional submanifold of solutions leading to a strict collision. The remainder of these Arnold tori has oscillating orbits of the second type and the measure analysis shows that the corresponding subset has a positive measure (Fejoz,1999, p. 76). 

The domain of oscillating orbits of the second type corresponds to the non-resonant orbits (Arnold tori) that satisfy ... 

The bounded motions. Almost all of them belong to an Arnold torus of quasi-periodic motions. 

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

The effect of sufficiently weak anharmonicities of the potential on this picture will be to distort the rectangle comprising the classical trajectories so that the motion occurs on a two-dimensional torus belonging to the three-dimensional constant energy subspace of the total four-dimensional phase space of the system [Arnold, 1978]. 

The procedure above allows us to prove only that there is an invariant torus close to the initial conditions of the Sun-Jupiter-Saturn system, not that the orbit of the system actually lies on a torus. Since we can not exclude the possibility of Arnold s diffusion, this is not enough to prove the perpetual stability of the orbit of the secular system. Therefore, we make a more accurate analysis in order to prove that the orbit is actually confined in a gap between two invariant tori. The procedure is illustrated in Figure 3. 

The paper is organized as follows in Section 2 and 3 we define the Fast Lyapunov Indicator and give some examples on the 2 dimensional standard map and on a Hamiltonian model. The special case of periodic orbits will be detailed in 4 and thanks to a model of linear elliptic rotation we will be able to recover the structure of the phase space in the vicinity of a noble torus. The use of the FLI for detecting the transition between the stable Nekhoroshev regime to the diffusive Chirikov s one will be recalled in Section 5. In 6 and 7 we will make use of the FLI results for the detection of the Arnold s diffusion. 

The dynamics of such systems is described by the Kolmogorov-Arnold-Moser theory of nearly integrable conservative dynamical systems (see e.g. Ott (1993)). For e = 0 the fluid elements move along the streamlines and the trajectories in the phase space form tubes parallel to the time axis. Due to the periodicity in the temporal direction these tubes form tori that fill the whole phase space and are invariant surfaces for the motion of the fluid elements. Each torus... 

Although in a weakly time-dependent flow all resonant tori disappear together with some of the nearly resonant tori around them, the Kolmogorov-Arnold-Moser theorem ensures that infinitely many invariant surfaces survive a small perturbation. For sufficiently small e the remaining invariant surfaces formed by quasiperiodic orbits, so called KAM tori, still occupy a non-zero volume of the phase space. The condition for a torus to survive a given perturbation is that its rotation number should be sufficiently far from any rational number so that the inequality... 

The first task is to construct semiclassical wave function in the classically allowed region. As is well known, the KAM torus exists according to the Kolmogorov-Arnold-Moser (KAM) [57,58] and this integral system can be quantized by the Einstein-Brillouin-Keller (EBK) quantization rule [58] as... 

---