A Reduced Order Modeling Framework for Strongly Perturbed Nonlinear Dynamical Systems Near Arbitrary Trajectory Sets

Abstract

A reduced order modeling strategy is proposed that can accurately capture the behavior of strongly perturbed nonlinear dynamical systems, i.e., those that are subject to general, large magnitude inputs. In contrast to standard variational approaches which consider dynamics in the neighborhood of a single reference trajectory, the proposed methodology augments the dynamics with an additional variable that continuously selects from a family of reference trajectories in order to limit truncation errors that result from neglected nonlinear terms. Provided the reference trajectories contract sufficiently rapidly in some directions, a reduced order set of equations can be obtained by choosing an appropriate coordinate system. Direct numerical approaches for computation of the necessary terms of the associated reduced order equations are provided. Crucially, because the proposed reduction strategy does not require the existence of a persistent, stable fixed point or periodic orbit, it can be implemented in situations where external inputs cause the dynamics to transition through a bifurcation. Two examples with relevance to neural control are provided. In the first, the proposed reduction strategy allows for the formulation of a numerically tractable control problem to identify energy-optimal inputs that eliminate tonic firing in a neural model. In the second example, the proposed reduction framework accurately captures successive transitions between quiescence and tonic firing across the boundary of a saddle-node on invariant circle bifurcation.