ManyLands: A Journey Across 4D Phase Space of Trajectories

Abstract

Mathematical models of ordinary differential equations are used to describe and understand biological phenomena. These models are dynamical systems that often describe the time evolution of more than three variables, i.e., their dynamics take place in a multi-dimensional space, called the phase space. Currently, mathematical domain scientists use plots of typical trajectories in the phase space to analyze the qualitative behavior of dynamical systems. These plots are called phase portraits and they perform well for 2D and 3D dynamical systems. However, for 4D, the visual exploration of trajectories becomes challenging, as simple subspace juxtaposition is not sufficient. We propose ManyLands to support mathematical domain scientists in analyzing 4D models of biological systems. By describing the subspaces as Lands, we accompany domain scientists along a continuous journey through 4D HyperLand, 3D SpaceLand, and 2D FlatLand, using seamless transitions. The Lands are also linked to 1D TimeLines. We offer an additional dissected view of trajectories that relies on small-multiple compass-alike pictograms for easy navigation across subspaces and trajectory segments of interest. We show three use cases of 4D dynamical systems from cell biology and biochemistry. An informal evaluation with mathematical experts confirmed that ManyLands helps them to visualize and analyze complex 4D dynamics, while facilitating mathematical experiments and simulations.