Haugaard Asger M
University of Oxford, Medical sciences division, Oxford, OX3 9DU, United Kingdom.
PLoS One. 2017 Jun 23;12(6):e0179507. doi: 10.1371/journal.pone.0179507. eCollection 2017.
Predicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to discretised manifolds of low dimension (low-D) embedded in high-dimensional (high-D) phase space. As a central feature, the method exploits that strange attractor dimension is generally much smaller than parent system phase space dimension. This is important, since the computational cost associated with discretised manifolds depends exponentially on their dimension. Thus, systems that would otherwise be associated with tremendous computational challenges, can be tackled on a laptop. As a test, bounding manifolds are calculated for high-D modifications of the canonical Duffing system. Parameters can be set such that the bounding manifold displays harmonic behaviour even if the underlying system is chaotic. Thus, solving for one post-transient forcing cycle of the bounding manifold predicts the extrema of the underlying chaotic problem indefinitely.
预测高维相空间中混沌系统的极值仍然是一个挑战。到目前为止，能够给出长期有效的极值的方法仅限于少数变量的模型。在此，提出了一种方法，该方法将混沌系统的极值视为属于嵌入在高维相空间中的低维离散流形。作为一个核心特征，该方法利用了奇异吸引子维度通常远小于母系统相空间维度这一特性。这一点很重要，因为与离散流形相关的计算成本与其维度呈指数关系。因此，那些原本会带来巨大计算挑战的系统，在笔记本电脑上就能处理。作为一个测试，为规范杜芬系统的高维变体计算了边界流形。可以设置参数，使得即使基础系统是混沌的，边界流形也能显示出谐波行为。因此，求解边界流形的一个瞬态后强迫周期就能无限期地预测基础混沌问题的极值。
