Connections between the modes of a nonlinear dynamical system on a manifold

An approach based on modal analysis is developed to describe the state space of a nonlinear dynamical system using finite-dimensional linear systems. This strategy provides meta-reduced-order modeling (meta-ROM) consisting of the modes, which are the linear combinations of extra-decomposed modes (meta-modes). We confirm that, in a broad range of parameter spaces, a meta-ROM can yield appropriate solutions for a simple nonlinear equation, in which the conventional ROM shows unphysical solutions. The meta-modal analysis on a manifold provides an insight into the description of the state space in low dimensional space. This interpretation leads to the description of nonlinear dynamics in terms of intrinsic geometry rather than conventional extrinsic geometry.