Exploring Chaotic Motion with Short-Time Expansion Rates

We present a new method for identifying the regions on a chaotic attractor that are locally more stable and hence potentially more predictable than other regions. To do this, we construct in each neighborhood of a chaotic attractor an independent coordinate system in which one axis is carefully aligned with the local flow direction and the remaining axes are aligned with the other dynamical directions. This creates a moving reference frame that evolves along a given trajectory, but is independent in the sense that its axes are determined by the attractor's local dynamical geometry and not by parametric properties of the trajectory itself. The novelty of our technique lies in its ability to consider the local dynamics of chaotic systems, while being robust to both noise and to any nonlinearities in the governing equations. We demonstrate our method with several classic chaotic, hyperchaotic, and conservative dynamical systems.