Searching chaotic saddles in high dimensions

Abstract

We propose new methods to numerically approximate non-attracting sets governing transiently-chaotic systems. Trajectories starting in a vicinity $\Omega$ of these sets escape $\Omega$ in a finite time $\tau$ and the problem is to find initial conditions $\mathbf{x}\in \Omega$ with increasingly large $\tau =\tau (\mathbf{x})$. We search points ${\mathbf{x}}^{\prime }$ with $\tau ({\mathbf{x}}^{\prime })>\tau (\mathbf{x})$ in a {\it search domain} in $\Omega$. Our first method considers a search domain with size that decreases exponentially in $\tau$, with an exponent proportional to the largest Lyapunov exponent ${\lambda }_1$. Our second method considers anisotropic search domains in the {\it tangent} unstable manifold, where each direction scale as the inverse of the corresponding {\it expanding} singular value of the Jacobian matrix of the iterated map. We show that both methods outperform the state-of-the-art {\it Stagger-and-Step} method (Sweet, Nusse, and York, Phys. Rev. Lett. {\bf 86}, 2261, 2001) but that only the anisotropic method achieves an efficiency independent of $\tau$ for the case of high-dimensional systems with multiple positive Lyapunov exponents. We perform simulations in a chain of coupled Hénon maps in up to 24 dimensions ($12$ positive Lyapunov exponents). This suggests the possibility of characterizing also non-attracting sets in spatio-temporal systems.

6 pages, 6 figures

References in corpus (2)

• Precision shooting: Sampling long transition pathways
• Efficiency of Monte Carlo Sampling in Chaotic Systems