自适应相位-振幅坐标框架下非线性振动的降阶表征
Reduced Order Characterization of Nonlinear Oscillations Using an Adaptive Phase-Amplitude Coordinate Framework
摘要：
我们提出了一个通用的策略，显示高度非线性振动系统的降阶建模。通过考虑一个连续的家庭强迫周期轨道定义在一个稳定的不动点，并随后利用相位振幅为基础的减少策略，我们到达一个低阶模型，能够准确地捕捉非线性振动所造成的任意外部输入。在振荡很小的限制下，系统动力学放松到从局部线性化获得的那些，即，这可以用线性本征模完全描述。对于较大振幅的振荡，可以根据少量非线性模式的动力学来理解其行为。我们说明了所提出的策略，在各种例子中产生的结果，大大优于使用标准的线性化为基础的技术。
英文摘要：
We propose a general strategy for reduced order modeling of systems that display highly nonlinear oscillations. By considering a continuous family of forced periodic orbits defined in relation to a stable fixed point and subsequently leveraging phase-amplitude-based reduction strategies, we arrive at a low order model capable of accurately capturing nonlinear oscillations resulting from arbitrary external inputs. In the limit that oscillations are small, the system dynamics relax to those obtained from local linearization, i.e.,~that can be fully described using linear eigenmodes. For larger amplitude oscillations, the behavior can be understood in terms of the dynamics of a small number of nonlinear modes. We illustrate the proposed strategy in a variety of examples yielding results that are substantially better than those obtained using standard linearization-based techniques.
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