A New Model Reduction Method for Nonlinear Dynamical Systems Using Singular PDE Theory

Summary

In the present research study a new approach to the problem of modelreduction for nonlinear dynamical systems is proposed. The formulation of the problem is conveniently realized through a system of singular quasi-linear invariance PDEs, and an explicit set of conditions for solvability is derived. In particular, within the class of real analytic solutions, the aforementioned set of conditions is shown to guarantee the existence and uniqueness of a locally analytic solution, which is then proven to represent the slow invariant manifold of the nonlinear dynamical system under consideration. As a result, an exact reduced-order model for the nonlinear system dynamics is obtained through the restriction of the original system dynamics on the aforementioned slow manifold. The local analyticity property of the solution’s graph that corresponds to the system’s slow invariant manifold enables the development of a series solution method, which allows the polynomial approximation of the “slow” system dynamics on the slow manifold up to the desired degree of accuracy.