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We propose a unified approach to nonlinear modal analysis in dissipativeoscillatory systems. This approach eliminates conflicting definitions, coversboth autonomous and time-dependent systems, and provides exact mathematicalexistence, uniqueness and robustness results. In this setting, a nonlinearnormal mode (NNM) is a set filled with small-amplitude recurrent motions: afixed point, a periodic orbit or the closure of a quasiperiodic orbit. Incontrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to aNNM, serving as the smoothest nonlinear continuation of a spectral subspace ofthe linearized system along the NNM. The existence and uniqueness of SSMs turnsout to depend on a spectral quotient computed from the real part of thespectrum of the linearized system. This quotient may well be large even forsmall dissipation, thus the inclusion of damping is essential for firmconclusions about NNMs, SSMs and the reduced-order models they yield.

Nonlinear normal modes and spectral submanifolds: Existence, uniqueness and use in model reduction