The Path towards a Longer Life: on Invariant Sets and the Escape Time Landscape

Unstable invariant sets are important to understand mechanisms behind many dynamically important phenomenon such as chaotic transients which can be physically relevant in experiments. However, unstable invariant sets are nontrivial to find computationally. Previous techniques such as the PIM triple method [Nusse & Yorke, 1989] and simplex method variant [Moresco & Dawson, 1999], and even the step-and-stagger method [Sweet et al., 2001] have computationally inherent dimension limitations. In the current study, we explicitly investigate the landscape of an invariant set, which leads us to a simple gradient search algorithm to construct points close to the invariant set. While the calculation of the necessary derivatives can be computationally very expensive, the methods of our algorithm are not as dimension dependant as the previous techniques, as we show by examples such as the two-dimensional instability example from [Sweet et al., 2001] followed by a four-dimensional instability example, and then a nine-dimensional flow from the Yoshida equations, with a two-dimensional instability.