Quasi-periodic function

The function cri appearing in Eq. (83) is an even quasi-periodic function (while cr, another Weierstrass function, is odd) defined by the relation [14]... 

The complete Lame spheroconal harmonic quasi-periodic functions, satisfying the respective boundary conditions in Eq. (91), involve matching the corresponding factors of Table 4.4 and their counterparts. For the boundary... 

The Lame quasi-periodic functions are common eigenfunctions of fhe operators U and H as discussed in Sections 2.2 and 2.5. In Refs. [1] and [8], we chose the notation of G for the latter in order to emphasize that it represents the geometry of the spheroconal elliptical cone confinement, in contrast with its dynamical character for the rotations of asymmetric molecules. The eigenvalues and /i are numerically found to satisfy the relationships... 

The stress needed to move a dislocation line in a glassy medium is expected to be the amount needed to overcome the maximum barrier to the motion less a stress concentration factor that depends on the shape of the line. The macro-scopic behavior suggests that this factor is not large, so it will be assumed to be unity. The barrier is quasi-periodic where the quasi-period is the average mesh size, A of the glassy structure. The resistive stress, initially zero, rises with displacement to a maximum and then declines to zero. Since this happens at a dislocation line, the maximum lies at about A/4. The initial rise can be described by means of a shear modulus, G, which starts at its maximum value, G0, and then declines to zero at A/4. A simple function that describes this is, G = G0 cos (4jix/A) where x is the displacement of the dislocation line. The resistive force is then approximately G(x) A2, and the resistive energy, U, is ... 

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

Following a fast Fourier transform of the data, the power spectrum shows the power (the Fourier transform squared) as a function of frequency. Random and chaotic data sets fail to demonstrate a dominant frequency. Periodic or quasi-periodic data sets will show one or more dominant frequencies [37]. 

In reaction dynamics, NHIMs with saddles would lose normal hyperbolicity as the energy of the vibrational modes increases at saddles. This is shown schematically in Fig. 32. Here, a saddle X of the potential function is displayed with its NHIM above in the phase space. When the reaction takes place with only a small amount of the energy in the vibrational modes, orbits go over the saddle where the vibrational motions are quasi-periodic. In Fig. 32, this is shown by the dotted arrow with tori on the NHIM. As the energy of the vibrational modes increases, however, orbits go over the saddle where the vibrational motions are chaotic because of the coupling among the vibrational modes. In Fig. 32, this is shown by the solid arrow with chaos (shown by the wavy line) on the NHIM. If the Lyapunov exponents of these chaotic motions become larger than those of the normal directions, the condition of normal hyperbolicity breaks down. 

It is seen that 0 t) has a very simple form. It is the sum over a countable set of periodic functions with coeflficients that vanish rapidly enough so that the sum over the coeflScients is 1. Sums of this type have been studied intensively in the mathematical hterature. Functions such as 6 t) are called quasi-periodic and it is known that they behave in a very orderly way which is reminiscent of simple periodic motion. 

The main novelty of representation (17) with respect to the quasi-periodic one consists in having replaced the constant amplitudes gk with amplitude functions gk(t). The key point of the result of Guzzo and Benettin (2001) are estimates (15), (16), which state that in the Nekhoro-shev regime the Fourier transform of such amplitude functions decays very fast with respect to the frequency. 

Figure 1 Individual mode energies, Hn from Eq. (14) for the coupled Morse oscillator system in Eq. (13) as functions of time measured in harmonic vibrational periods. Each figure represents a single trajectory. The total energy in each trajectory is D, the dissociation energy of a single mode, (a) Localization of the excitation. The two modes are not in nonlinear resonance, (b) Quasi-periodic exchange of the excitation. The two modes are in nonlinear resonance.

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