Lyapunov Time for 2D Lagrangian Visualization

Many concepts in computational flow visualization operate in the Lagrangian frame—they involve the integration of trajectories. A problem inherent to these approaches is the choice of an appropriate time length for the integration of these curves. While for some applications the choice of such a finite time length is straightforward, it represents in most other applications a parameter that needs to be explored and well-chosen. This becomes even more difficult in situations where different regions of the vector field require different time scopes. In this chapter, we introduce Lyapunov time for this purpose. Lyapunov time, originally defined for predictability purposes, represents the time over which a trajectory is predictable, i.e., not dominated by error. We employ this concept for steering the integration time in direct visualization by trajectories, and for derived representations such as line integral convolution and delocalized quantities. This not only provides significant visualizations related to time-dependent vector field topology, but at the same time incorporates uncertainty into trajectory-based visualization.