  We study cocycles (non-autonomous dynamical systems) satisfying a certain
squeezing condition with respect to the quadratic form of a bounded
self-adjoint operator acting in a Hilbert space. We prove that (under
additional assumptions) the orthogonal projector maps the fibres of some
invariant set, containing bounded trajectories, in a one-to-one manner onto the
negative subspace of the operator. This allows to reduce interesting dynamics
onto this invariant set, which in some cases can be considered as a kind of
inertial manifold for the cocycle. We consider applications of the reduction
principle for periodic cocycles. For such cocycles we give an extension of the
Massera second theorem, obtain the conditions for the existence of a Lyapunov
stable periodic trajectory and prove convergence-type results, which we apply
to study nonlinear periodic in time delayed-feedback equations posed in a
proper Hilbert space and parabolic problems with a nonlinear periodic in time
boundary control. The required operator is obtained as a solution to certain
operator inequalities with the use of the Yakubovich-Likhtarnikov frequency
theorem for $C_{0}$-semigroups and its properties are established from the
Lyapunov inequality and dichotomy of the linear part of the problem.
