Towards Exact Statistically Independent Nonlinear Normal Modes via the FPK Equation

The nonlinear extension of linear modal analysis is a problem that has received a great deal of attention in the literature. A recently proposed framework casts the problem of nonlinear modal analysis within a machine-learning framework that generates modes that are statistically independent from each other. Thus far, several techniques from machine learning have been applied to learn the nonlinear modal transformation, including a multinomial expansion and a neural-network model. Recent effort has been given to understand the nonlinear modal dynamics within the statistically independent framework from a theoretical perspective; of particular interest are ways that the nonlinear transformations might be specified exactly from the equations of motion. This paper uses the Fokker-Planck-Kolmogorov (FPK) equation to construct a transformation in the reduced single degree-of-freedom nonlinear case; in this way, the dynamics can be made to respect an arbitrary target distribution. It is furthermore shown that setting the target distribution to the Gaussian distribution corresponding to the underlying linear dynamics (whereby all of the nonlinear elements are removed) is not sufficient to produce global amplitude invariance in the modal transformation.