Periodic orbit simple

Currently available numerical results indicate that the one-dimensional heUum atom is completely chaotic. The best-known semiclassical quantization procedure for completely chaotic systems is Gutzwiller s trace formula (see Section 4.1.3), which is based on classical periodic orbits. Therefore we search for simple periodic orbits of the one-dimensional he-hum atom. Since a two-electron orbit is periodic if the orbits ni t), 0i t)) and (ri2(t), 2( )) of the first and second electron have a common period, the periodic orbits of the one-dimensional model can be labelled with two integers, m and n, which count the 27r-multiplicity of the angle variables 0i and 02 after completion of the orbit. Therefore, if for some periodic orbit... 

The properties of a few simple periodic orbits are displayed in Table 10.1. Column 1 indicates the orbit in m n notation, column 2 is its reduced symbolic code, column 3 lists the action 5 = Pi dxi of the first electron and column 4 lists the action Sn = P2 dx2 of the second electron. Column 5 displays the total action 5 of the orbit and column 6 lists the scaled traversal time r of the orbit. The natural traversal times T of the orbits are not listed since they are obtained trivially from (10.3.11) as T = 5/2. 

Table 10.1. Properties of a few simple periodic orbits. (Adapted from Bliimel and Reinhardt (1992).)...

Fig. 10.10 proves that a close connection exists between the classical mechanics and the quantum mechanics of the simple one-dimensional two-electron model. On the basis of the evidence provided by Fig. 10.10, there is no doubt that classical periodic orbits determine the structure of the level density in an essential way. The key element for establishing the one-to-one correspondence between the peaks in R and the actions of periodic orbits is the scaling relations (10.3.10). Similar relations hold for the real helium atom. Therefore, it should be possible to establish the same correspondence for the three-dimensional helium atom. First steps in this direction were taken by Ezra et al. (1991) and Richter (1991). 

If the omega limit set is particularly simple - a rest point or a periodic orbit - this gives information about the asymptotic behavior of the trajectory. An invariant set which is the omega limit set of a neighborhood of itself is called a (local) attractor. If (3.1) is two-dimensional then the following theorem is very useful, since it severely restricts the structure of possible attractors. 

Figure 3.1 illustrates the possibilities. Additionally, if a two-dimensional system has a periodic orbit then it must have a rest point inside that orbit. These simple facts (and their generalizations) play an important role in the analysis presented here. 

The studies of Wiesenfeld [28] and Lai et al. [43] on the classical dynamics of a one-electron atom in a sinusoidal external field provide a physically realistic example in which the presence of KAM tori surrounding stable periodic orbits leads to deviations from the generic behaviour characteristic of a hyperbolic scattering system as discussed in Sect. 2. Although this system (10) seems simple, further studies illuminating the mathematical structures behind the scattering process, e.g. calculation of the Liapunov exponents of the unstable trapped orbits and the fractal dimension of the trapped set, have yet to be performed. 

When 7 >> 1, as it is in our study of the Fibonacci sequence, then the FLI, i.e. the supremum of the norm of v behaves like 7 (0), i.e. like a since vy(0) b. We remark that if vx(0) = 0 then vy(0) = b. A question remains about the transitory regime in which the FLI grows linearly with time. Actually, in order to reach its maximum value, the vector v(t) has to visit the ellipse from the the semi-minor to the semimajor axis, i.e. has to rotate at least an angle of 90 degrees. We were able to reproduce very well the evolution of the FLI with time of the seven Fibonacci periodic orbits (Lega and Froeschle 2000), confirming the validity of the simple model for explaining the behavior of the FLI for periodic orbits. 

Measure of the size of the islands. In the light of the results obtained with the simple model, we can test if by computing the FLI for periodic orbits we can recover the geometric parameter 7 = a/b. 

Figure 9.19 Overview of the phase space and configuration space dynamics associated with the HCCH [JV = 22, l = 0] polyad. The top four plots are surfaces of section for four energies within the polyad. Only simple structures are found near the bottom (local bender) and top (counter-rotators) of the polyad. Chaos dominates at energies near the middle of the polyad, but two classes of stable trajectories coexist with chaos. The bottom four plots show the coordinate space motions of the left- and right-hand H-atoms that correspond to the two different periodic orbits (from Jacobson and Field, 2000b).

Depending on whether the representative point of the system iterates from the right branch/j(x ) to the horizontal branch/i(x ) immediately, or by making an excursion to the median branch /2(x ), the constrained periodic orbit will be of simple or complex nature. A simple CPO corresponds to a pattern of bursting of the type tt(/), while a complex CPO will correspond to the bursting type ir(/, k,l. ..) where the number of indices reflects the number of times the iteration will bring the system on to the median branch /2(x ) before the constrained periodic orbit ends by iteration towards the horizontal branch /i(x ). The number of... 

The equations of the piecewise linear map allow us to determine the domains of existence of simple, constrained periodic orbits with p peaks, tt(/ ), as a function of parameter a. The mode Tr(p) is obtained as soon as condition (4.7) is satisfied ... 

Figure 4.27 shows how the variation of parameter a influences the number of peaks of constrained periodic orbits of the simple type tt(p). 

Fig. 4.27. Domains of values of a for which constrained periodic orbits of the simple type ir(p) exist in the piecewise linear map defined by eqns (4.5). Parameter values for the map are those of fig. 4.26 (Decroly Goldbeter, 1987).

What happens in the intervals of a values that do not produce any constrained periodic orbit of the simple type The numerical simulation of eqns (4.5) shows that constrained periodic orbits of the complex type are then observed. Typical results of such simulations are represented in fig. 4.28, for ft = 7, Af = 11. The upper panel shows the successive values of X, obtained according to eqns (4.5), as a function of parameter a varying from 1.76 to 1.82. The number of values of x for a given value of a is equal to the total number of peaks in the pattern of bursting. Values marked a-g correspond, respectively, to the patterns ir(6,2), tt(6, 2, 6), 7r(6, 2, 5), it(6, 2, 4), Tr(6, 2, 3), ir(6, 2, 2) and ir(6, 3). These complex patterns of bursting nevertheless remain relatively simple when compared to those that occur in the separating intervals the existence of the latter, extremely complex patterns is reflected in the huge number of values of x obtained for certain values of a. 

As we have already seen in the previous section, finding adiabatic barriers and wells of the n-th quantal vibrational adiabatic potential surface for the y dependent Hamiltonian h(y) is equivalent semiclassica-lly to finding periodic orbits of h(y) with quantised action - (n+l/2)h if the periodic orbit is over a simple well potential. The time dependent coordinates and momenta of the (y dependent) periodic orbit are denoted r (t y), R (t y), Pr(t y), and PR(t y), and the period of the orbit is T (y). We thus find for each value of y a vibrationally adiabatic barrier or well at energy E (y), a stability frequency o)n(y) and effective mass M (uq) (cf. Eq. 27) for motion perpendicular to the... 

Note that in the n-dimensional case, where n > 4, other topological configurations of may be realized. Such saddle-node bifurcations will definitely lead the system out of the class of systems with simple dynamics. For example, it is shown in [139, 152] that a hyperbolic attractor of the Smale-Williams type may appear just after the disappearance of a saddle-node periodic orbit. ... 

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