Approximation of trajectories lying on a global attractor of a hyperbolic equation with exterior force rapidly oscillating in time

A quasilinear dissipative wave equation is considered for periodic boundary conditions with exterior force g(x, t/epsilon) rapidly oscillating in t. It is assumed in addition that, as (t)epsilon --> 0+, the-function g(x, t/epsilon) converges in the weak sense (in L-2,w(loc) (R, L-2 (T-n)) to a function (g) over bar (x) and the averaged-wave. equation (with exterior force (g) over bar (x)) has only finitely many stationary point's {z(i)(x), i= 1,..., N}, each of them hyperbolic. It is proved that the global attractor A(epsilon) of the original equation deviates in the energy norm from the global attractor A(0) of the averaged equation by a quantity Cepsilon(rho), where rho is described by an explicit formula. It is also shown that each piece of a trajectory u(epsilon)(t) of the original equation lying on A(epsilon) that corresponds to an interval of time-length Clog(1/epsilon) can be approximated to within C(1)epsilon(rho1) by means of finitely many pieces of trajectories lying on unstable manifolds M-u(z(i)) of the averaged equation, where an explicit expression for rho(1) is provided.