Failing parametrizations: what can go wrong when approximating spectral submanifolds

Abstract

Invariant manifolds provide useful insights into the behavior of nonlinear dynamical systems. For conservative vibration problems, Lyapunov subcenter manifolds constitute the nonlinear extension of spectral subspaces consisting of one or more modes of the linearized system. Conversely, spectral submanifolds represent the spectral dynamics of non-conservative, nonlinear problems. While finding global invariant manifolds remains a challenge, approximations thereof can be simple to acquire and still provide an effective framework for analyzing a wide variety of problems near equilibrium solutions. This approach has been successfully employed to study both the behavior of autonomous systems and the effects of non-autonomous forcing. The current computation strategies rely on a parametrization of the invariant manifold and the reduced dynamics thereon via truncated power series. While this leads to efficient recursive algorithms, the problem itself is ambiguous, since it permits the use of various approaches for constructing the reduced system to which the invariant manifold is conjugated. Although this ambiguity is well known, it is rarely discussed and usually resolved by an ad hoc choice of method, the effects of which are mostly neglected. In this contribution, we first analyze the performance of three popular approaches for constructing the conjugate system: the graph style parametrization, the normal form parametrization, and the normal form parametrization for “near resonances.” We then show that none of them is always superior to the others and discuss the potential benefits of tailoring the parametrization to the analyzed system. As a means for illustrating the latter, we introduce an alternative strategy for constructing the reduced dynamics and apply it to two examples from the literature, which results in a significantly improved approximation quality.