The Path Towards a Longer Life: On Invariant Sets and the Escape Time Landscape

Abstract

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.

Edge of Chaos in a Parallel Shear Flow

“…2). This behavior can be considered a typical ''lifetime landscape'' [3,4,17] for a chaotic saddle. As illustrated in Fig.…”

Importance sampling of rare events in chaotic systems

“…Numerical investigations of transiently-chaotic systems are computationally difficult because most trajectories quickly escape the vicinity of the chaotic saddle (on which the chaotic dynamics is properly defined) [11,12,45]. with the state x, shows an intricate landscape with multiple local and global maxima.…”

“…In computations of the FTLE in closed systems we are interested in states with increasing N = t o and on the tails of P (E). These problems share two distinct computational problems: find rare states [10][11][12]16] and sample rare states [18,19,24,25], which can be formalized as follows:…”

“…in references [11,12,16,18,65]. Specifically, these formulas were derived to guarantee equation (34), which dictates how different E [E(x )|x] has to be from E(x) to guarantee a constant acceptance.…”

“…The importance of rare trajectories in chaotic systems, including methods designed to find and sample them, have been studied through different perspectives [9][10][11][12][13][14][15][16][17][18][19][20]. For example, the method stagger and dagger, used to find longliving trajectories of transiently chaotic systems, has been an important tool to characterize chaotic saddles [10,11].…”