Nonlinear dynamics in problems of stability of complex media

The problem of stability of nonlinear viscoelastic bodies with respect to finite perturbations is considered in this article. The analysis of the basic process of deformation of a viscoelastic medium is reduced to solution of a nonlinear boundary value problem with variable coefficients. Solutions for perturbations of displacements are in the form of series in eigenfunctions. Using the principle of possible displacements the question on stability of the ground state of a variational nonlinear problem is reduced to investigation of stability of the zero solution of an infinite system of ordinary differential equations with constant coefficients. For the resulting system of equations we construct a function, which under certain restrictions on the initial perturbations is a Lyapunov function. The dimension of the strange attractor of the dynamical system which allows to limit the number of terms in a Bubnov-Galerkin set is found.