非线性正规模和谱子流形：存在性、唯一性及其在模型降阶中的应用
Nonlinear normal modes and spectral submanifolds: Existence, uniqueness and use in model reduction
摘要：
提出了一种耗散振动系统非线性模态分析的统一方法。这种方法消除了相互冲突的定义，涵盖了自治和时间依赖的系统，并提供精确的数学存在性，唯一性和鲁棒性的结果。在这种情况下，一个非线性正常模式（NNM）是一组充满小振幅回复运动：一个不动点，周期轨道或准周期轨道的封闭。相比之下，谱子流形（SSM）是NNM的不变流形渐近，作为线性化系统的谱子空间沿NNM的最光滑的非线性延拓。SSM的存在性和唯一性取决于从线性化系统的谱的实部计算的谱商。这个商可能很大，即使是小的耗散，因此，包括阻尼是必不可少的关于NNMs，SSM和降阶模型，他们产生的坚定的结论。
英文摘要：
We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems, and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation, thus the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield.
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