Dimensionality Reduction of Dynamical Systems with Parameters: A Geometric Approach

Abstract

A method for obtaining a low-dimensional description of a family of attractors produced by continuous-time nonlinear dynamical systems with static parameters is developed. A geometric approach is taken, which allows for the reduction of general state space manifolds, not restricted to $\mathbb{R}^n$. An existing secant-based projection method, utilizing optimization over Grassmann manifolds, is extended for use with multiple parameter values and data sets. A family of reduced vector fields is obtained, with parameterization by the original parameter space, that reproduces the projected attractors and their dynamics in a low-dimensional space. We illustrate the method with several examples. The Rössler system demonstrates the accurate reproduction of period-doubling bifurcations; a forced damped double pendulum demonstrates a nonautonomous system with angular variables; and the Brusselator demonstrates application to a high-dimensional system.