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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 x is an element of Omega with increasingly large tau = tau(x). We search points x' with tau(x') > tau(x) in a 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 tangent unstable manifold, where each direction scales as the inverse of the corresponding expanding singular value of the Jacobian matrix of the iterated map. We show that both methods outperform the state-of-the-art Stagger-and-Step method [Sweet et al., Phys. Rev. Lett. 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 Henon maps in up to 24 dimensions (12 positive Lyapunov exponents). This suggests the possibility of characterizing also non-attracting sets in spatio-temporal systems. Published by AIP Publishing.