Summary: 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 \(\boldsymbol{x} \in \Omega\) with increasingly large \(\tau = \tau(\boldsymbol{x})\). We search points \(\boldsymbol{x}^\prime\) with \(\tau(\boldsymbol{x}^\prime) > \tau(\boldsymbol{x})\) 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 [D. Sweet et al., Phys. Rev. Lett. 86, No. 11, 2261 (2001; doi:10.1103/PhysRevLett.86.2261)] 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.{
©2016 American Institute of Physics}