Existence of Spectral Submanifolds in Time Delay Systems

Spectral submanifolds (SSMs) are invariant manifolds of a dynamical system, defined by the property of being tangent to a spectral subspace of the linearized dynamics at a steady state. We show existence, along with certain desirable properties such as smoothness, attractivity and conditional uniqueness, of SSMs associated to a large class of spectral subspaces in time delay systems. Building on these results, we generalize the criteria for existence of inertial manifolds – defined as globally exponentially attracting Lipschitz invariant manifolds of finite dimension – and show that they need not have dimension equal to that of the physical configuration, in contrast to previous accounts. We then demonstrate the applicability of these results on a few simple examples.