Sala M, Leitão J C, Altmann E G
Max Planck Institute for the Physics of Complex Systems, Nöthnizer Straße 38, 01187 Dresden, Germany.
Chaos. 2016 Dec;26(12):123124. doi: 10.1063/1.4973235.
We propose new methods to numerically approximate non-attracting sets governing transiently chaotic systems. Trajectories starting in a vicinity Ω of these sets escape Ω in a finite time τ and the problem is to find initial conditions x∈Ω with increasingly large τ=τ(x). We search points x' with τ(x')>τ(x) in a search domain in Ω. Our first method considers a search domain with size that decreases exponentially in τ, with an exponent proportional to the largest Lyapunov exponent λ. 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 τ 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.
我们提出了新的方法来数值逼近控制瞬态混沌系统的非吸引集。从这些集合的邻域Ω开始的轨迹会在有限时间τ内逃离Ω，问题在于找到初始条件x∈Ω，使得τ = τ(x)越来越大。我们在Ω的搜索域中寻找τ(x') > τ(x)的点x'。我们的第一种方法考虑一个搜索域，其大小随τ呈指数下降，指数与最大李雅普诺夫指数λ成正比。我们的第二种方法考虑在切向不稳定流形中的各向异性搜索域，其中每个方向的缩放比例与迭代映射的雅可比矩阵相应的扩张奇异值的倒数成正比。我们表明这两种方法都优于当前最先进的交错步长法[斯威特等人，《物理评论快报》86, 2261 (2001)]，但对于具有多个正李雅普诺夫指数的高维系统，只有各向异性方法实现了与τ无关的效率。我们在高达24维（12个正李雅普诺夫指数）的耦合亨农映射链中进行了模拟。这表明在时空系统中表征非吸引集也是有可能的。
