Two-dimensional torus

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 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. 

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. 

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... 

Chaos does not occur as long as the torus attraaor is stable. As a parameter of the system is varied, however, this attractor may go through a sequence of transformations that eventually render it unstable and lead to the possibility of chaotic behavior. An early suggestion for how this happens arose in the context of turbulent fluid flow and involved a cascade of Hopf bifurcations, each of which generate additional independent frequencies. Each additional frequency corresponds to an additional dimension in phase space the associated attractors are correspondingly higher dimensional tori so that, for example, two independent frequencies correspond to a two-dimensional torus (7 ), whereas three independent frequencies would correspond to a three-dimensional torus (T ). The Landau theory suggested that a cascade of Hopf bifurcations eventually accumulates at a particular value of the bifurcation parameter, at which point an infinity of modes becomes available to the system this would then correspond to chaos (i.e., turbulence). 

Whereas the fractal torus is difficult to distinguish from the wrinkled torus, the broken torus (stage 4) is immediately recognizable from its surface of section. The transition from wrinkled to fractal torus can, however, be clearly seen in the associated circle map. The circle map develops an inflection point (see Figure 34) at the transition from wrinkled to fractal torus. The existence of an inflection point means that the circle map is no longer invertible, that is, the circle map cannot be derived from a true two-dimensional torus. It also means that chaotic dynamics are now possible. The transition from stage 2 to stage 3 heralds the death of the two-dimensional torus and the transition to the possibility of chaotic dynamics. 

Consider a geodesic flow of a flat two-dimensional torus, that is, a torus with a locally Euclidean metric. This flow is integrable in the class of Bott integrals and obviously has no closed stable trajectories. By virtue of Proposition 2.1.2, we must have rank i(Q) 2. Indeed, the nonsingular surfaces Q are diffeomorphic here to a three-dimensional torus T, for which Hi T, Z) = Z 0 Z 0 Z. 

The number k of such circles is precisely equal to the rank of the one-dimensional integer-valued homology group Hi(My2), Moreover, this group is free (in the case of a non-empty boundary) and is isomorphic to Z 0 l(k times) For instance, removing one disk from a two-dimensional torus, we obtain with one hole homotopically equivalent to a bouquet of two circles, that is, in our example k = 2 and x M) = 1 - 2 = -1 (Fig. 77). 

Of special interest in nonlinear dynamics are the flows on a two-dimensional torus. We consider the systems on a torus which have no equilibrium states... 

It is important to distinguish the P , P and P-stable trajectories from each other. Indeed, consider the example from Sec. 1.2 of a system on a two-dimensional torus which possesses an equilibrium state with a P -trajectory which is a-limiting to the equilibrium state and a P -trajectory which is cj-limiting to it all other trajectories on the torus are Poisson-stable, and cover it densely. 

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... 

Fig. 11.7.5. The typical scenario of the breakdown of a two-dimensional torus due to a loss of its smoothness.

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... 

The Cherry flow is a flow on a two-dimensional torus with two equilibrium states a saddle and an unstable node both unstable separatrices are stable one stable separatrix is a-limit to a node and the other lies in the closure of the unstable separatrices and it is P -stable [see Fig. 13.7.4(a)]. The closure of the unstable separatrices is a quasiminimal set which contains the saddle O and a continuum of unclosed P-stable trajectories. The rotation number for such flows is defined in the same way as for flows on a torus without equilibrium states. Since there is no periodic orbits in a Cherry flow. 

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