Phase Space Projection of Dynamical Systems

Dynamical systems are commonly used to describe the state of time‐dependent systems. In many engineering and control problems, the state space is high‐dimensional making it difficult to analyze and visualize the behavior of the system for varying input conditions. We present a novel dimensionality reduction technique that is tailored to high‐dimensional dynamical systems. In contrast to standard general purpose dimensionality reduction algorithms, we use energy minimization to preserve properties of the flow in the high‐dimensional space. Once the projection operator is optimized, further high‐dimensional trajectories are projected easily. Our 3D projection maintains a number of useful flow properties, such as critical points and flow maps, and is optimized to match geometric characteristics of the high‐dimensional input, as well as optional user constraints. We apply our method to trajectories traced in the phase spaces of second‐order dynamical systems, including finite‐sized objects in fluids, the circular restricted three‐body problem and a damped double pendulum. We compare the projections with standard visualization techniques, such as PCA, t‐SNE and UMAP, and visualize the dynamical systems with multiple coordinated views interactively, featuring a spatial embedding, projection to subspaces, our dimensionality reduction and a seed point exploration tool.