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.