The deterministic core of stochastically perturbed nonlinear mechanical systems

Invariant manifolds, such as normally hyperbolic invariant manifolds and spectral submanifolds, are the key to understanding the dynamical behaviour of many nonlinear mechanical systems and serve as natural candidates for model-order reduction. While numerous related invariant manifold results are available for unforced and periodically forced nonlinear mechanical systems, their applicability to random external forcing remains to be established. Here, we clarify the continued role of deterministic invariant manifolds, more specifically normally hyperbolic invariant manifolds and spectral submanifolds under the addition of small white noise excitation. Subsequently, we demonstrate our results on several mechanical systems.