Phase space torus

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

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 trajectories of dissipative dynamic systems, in the long run, are confined in a subset of the phase space, which is called an attractor [32], i.e., the set of points in phase space where the trajectories converge. An attractor is usually an object of lower dimension than the entire phase space (a point, a circle, a torus, etc.). For example, a multidimensional phase space may have a point attractor (dimension 0), which means that the asymptotic behavior of the system is an equilibrium point, or a limit cycle (dimension 1), which corresponds to periodic behavior, i.e., an oscillation. Schematic representations for the point, the limit cycle, and the torus attractors, are depicted in Figure 3.2. The point attractor is pictured on the left regardless of the initial conditions, the system ends up in the same equilibrium point. In the middle, a limit cycle is shown the system always ends up doing a specific oscillation. The torus attractor on the right is the 2-dimensional equivalent of a circle. In fact, a circle can be called a 1-torus,... 

The KAM theorem demonstrates the existence of KAM tori when the perturbations to the motion are small. What happens when a nearly integrable Hamiltonian is strongly perturbed For example, with increasing perturbation strength, what is the last KAM torus to be destroyed and how should we characterize the phase space structures when all KAM tori are destroyed Using simple dynamical mapping systems, which can be regarded as Poincare maps in Hamiltonian systems with two DOFs, MacKay, Meiss, and Percival [8,9] and Bensimon and Kadanoff [10] showed that the most robust KAM curve... 

For easiness of computation, we impose a periodic boundary condition for p as well as 0 the phase space of the corresponding classical system becomes a two-dimensional torus [22,23]. In this case, Planck s constant is given by h = 2kM/tN, where p = Mn defines the periodic boundaries in the momentum space, and N is the number of discrete points describing 0 and p. In the actual calculations, we set x = 1. 

In this case the system is called integrable. From the first equation we have /, = const, for all i that is, /, are constants of motion. Thus / , (/) = 0//o/8/,- are also constants, and we have

phase space of the system is on a regular orbit (torus). 

The dimension of the phase space is 2N, as we have N / s and N (p s. If we still have global conservation laws (such as conservation of energy), the net dimension of the phase space is (/phasespace = 2N — (number of conservation laws). The problem is whether a KAM torus can divide the phase space (or energy surface) into two disjoint parts or not. The necessary condition for global drift of chaotic motion is... 

Besides the plane and the cylinder, another important two-dimensional phase space is the torus. It is the natural phase space for systems of the form... 

Thus, the orbits in the domain Q x Tn of phase space lie on invariant tori parameterized by the action variables h,..., /n, and the motion on each torus is a Kronecker flow with frequencies wi(/),..., ojn(I). 

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... 

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). 

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... 

If the variables are not separable, but the system nonetheless possesses N single-valued and independent integrals of motion, then motion takes place on the surface of an TV-dimensional torus in phase space. Within this surface, one can define N topologically distinct closed contours (labelled Ck, with k = 1, ...,TV), which are irreducible, i.e. cannot be turned into each other by continuous deformations. Examples are shown in fig. 10.2. 

Second, calculations can be performed in a semiclassical regime, and the results plotted on a Poincare section in action-angle (I, 0) coordinates. Such diagrams may seem complicated (see figs. 10.16 and 10.17), but are at least in principle readily understood a near-horizontal line across the (1,0) plot corresponds to a torus in ordinary phase space. When periodically extended in the time coordinate, each line corresponds to a vortex tube embedded in the extended phase space of the periodically... 

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... 

As discussed above (section 4.3.2) these local-mode states are not the eigenstates for the system, but superposition states. However, since the classical motion is quasiperiodic for these local mode states (i.e., the state is a torus in the phase space), the system is trapped in the initially excited local-mode state and the quantum periodic oscillation between the n,m) and (m,n) local mode states is not observed classically. Thus, classical mechanics severely underestimates the rate of energy transfer. 

First, consider the case of trapped motion within a single isomer. The phase space of 2 is (always) an ellipse, which has the same topology as a onedimensional sphere (which a mathematician would name S ). However, the phase space of is also elliptical and has the same topology (S ). The topology of the two-dimensional phase-space surface on which the dynamics lies is the Cartesian product of these two, which is a two-dimensional torus, or a phase-space doughnut (T = SI X The toroidal geometry is shown in... 

Figure 8 Representative phase space surfaces for uncoupled two-state isomerization in two degrees of freedom. A reactive torus labeled spans both isomers, while...

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