Notice that in the limiting model (6) the corresponding set of amplitude equations is reduced just to the single pendulum-type equation frequently used in many applications:
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It is known that this equation can possess unstable solutions at small values ​​of and. p> Solutions to eqs. (7) can be found using iterative methods of slowly varying phases and amplitudes:
(10);,
where and are new unknown coordinates. p> By substituting this into eqs. (9), we obtain the first-order approximation equations
(11);,
where is the coefficient of the parametric excitation; is the generalized phase governed by the following differential equation
.
Equations (10) and (11), being of a Hamiltonian structure, possess the two evident first integrals
and,
which allows one to integrate the system analytically. At, there exist quasi-harmonic stationary solutions to eqs. (10), (11), as
,
which forms the boundaries in the space of system parameters within the first zone of the parametric instability.
From the physical viewpoint, one can see that the parametric excitation of bending waves appears as a degenerated case of nonlinear wave interactions. It means that the study of resonant properties in nonlinear elastic systems is of primary importance to understand the nature of dynamical instability, even considering free nonlinear oscillations.
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Normal forms
The linear subset of eqs. (0) describes a superposition of harmonic waves characterized by the dispersion relation
,
where refer the branches of the natural frequencies depending upon wave vectors. The spectrum of the wave vectors and the eigenfrequencies can be both continuous and discrete one that finally depends upon the boundary and initial conditions of the problem. The normalization of the first order, through a special invertible linear transform
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leads to the following linearly uncoupled equations
,
where the matrix is composed by-dimensional polarization eigenvectors defined by the characteristic equation
;
is the diagonal matrix of differential operators with eigenvalues; and are reverse matrices. p> The linearly uncoupled equations can be rewritten in an equivalent matrix form [5]
(12) and,
using the complex variables. Here is the unity matrix. Here is the-dimensional vector of nonlinear terms analytical at the origin. So, this can be presented as a series in, i. e.
,
where are the vectors of homogeneous polynomials of degree, e. g.
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Here and are some given differential operators. Together with the system (12), we consider the corresponding linearized subset
(13) and,
