Computing parameter-dependent invariant manifolds for data-free nonlinear model reduction

Abstract

Invariant manifolds provide a solid foundation for rigorous model reduction and analysis of complex dynamical behavior in high-dimensional nonlinear systems. Their existence and smooth parameter dependence are well established theoretically; yet the practical computation of parameter-dependent invariant manifolds remains challenging, particularly for systems with nontrivial equilibria, large state dimension, or limited access to governing equations. In this work, we develop efficient computational expressions for invariant manifolds and their reduced dynamics that depend smoothly on system parameters, without shifting coordinates or introducing parameters as auxiliary variables. This approach enables formal Taylor and Taylor–Fourier expansions of parameter-dependent equilibria, invariant manifolds, and reduced-order models directly from the governing equations. We show how our expressions can be implemented nonintrusively for systems with affine parameter dependence and up to cubic nonlinearities. We demonstrate the effectiveness of the method on a range of high-fidelity numerical models, where the resulting parameter-dependent reduced-order models enable the analysis of local bifurcations, including buckling, flutter, period-doubling, and internal and parametric resonances, at computational costs far below those of the full system. The framework provides a practical bridge between rigorous invariant manifold theory and large-scale engineering applications, with particular relevance to design and optimization.