A non-local reduction principle for cocycles in Hilbert spaces

TL;DR

This paper introduces a reduction principle for cocycles in Hilbert spaces, enabling the analysis of complex dynamics through invariant sets and extending classical theorems to nonlinear periodic systems.

Contribution

It develops a new reduction method for cocycles satisfying a squeezing condition, extending the Massera second theorem and analyzing nonlinear periodic delayed-feedback and boundary control problems.

Findings

Reduction of dynamics onto invariant sets as inertial manifolds

Extension of Massera's theorem for periodic cocycles

Conditions for Lyapunov stability and convergence in nonlinear systems

Abstract

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.