where H = [pi]/2(1-k), [PHI]([THETA]) - a

periodic function of the variable [THETA] with period 2[pi]/[sigma] has periodic solutions with the same period if

The almost

periodic function introduced seminally by Bohr in 1925 plays an important role in describing the phenomena that are similar to the periodic oscillations which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, and ecosphere.

Let [beta](t) = a + b cos(4t), [gamma](t) = [[gamma].sub.0] with a, b, [[gamma].sub.0] [member of] R and [alpha](t), [chi](t) be [C.sup.2]

periodic function with the period [pi].

Let [B.sup.[alpha].sub.0](x) be a

periodic function of period 1.

As a function that controls the periodic time-variability of the model q(k) it is assumed following

periodic functionTo describe the time dependence of the winding angle we must look for a

periodic function, because motion of the point is periodic to a good approximation.

the r-th derivative of a

periodic function f is a particular case of the ([psi], [beta])-derivative for the sequence ([[psi].sub.k]) = ([k.sup.-r]) and [beta] = r.

We stress the fact that Corollary 3.6 applies to an arbitrary nonconstant

periodic function of class [C.sup.2] provided that its period [T.sub.[OMEGA]] is very large and its displacement from a constant value k > 0 is very small.

To simplify the calculation we shall assume that the refractive index is r(z) = [r.sub.o](1 + q(z)) with q(z), a sectionally constant

periodic function defined as

Since the vector [??](t) = [??](t + T) is a

periodic function of time, in the steady state (at s = const) the solution of the system of equation (3) is T-periodic dependences of the vector [??](t) = [??](t + T) components.

Objective: The origin of Harmonic Analysis goes back to the study of the heat diffusion, modeled by a differential equation, and the claim made by Fourier that every

periodic function can be represented as a series of sines and cosines.

where [phi] = [([[phi].sub.1], [[phi].sub.2], ..., [[phi].sub.n]).sup.T] is an arbitrary almost

periodic function. Based on system (37) and by using the exponential dichotomy of linear system, the following mapping was established: