Gottwald Georg A, Melbourne Ian
School of Mathematics and Statistics, University of Sydney, Sydney 2006 NSW, Australia.
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom.
Chaos. 2014 Jun;24(2):024403. doi: 10.1063/1.4868984.
Dynamics arising persistently in smooth dynamical systems ranges from regular dynamics (periodic, quasiperiodic) to strongly chaotic dynamics (Anosov, uniformly hyperbolic, nonuniformly hyperbolic modelled by Young towers). The latter include many classical examples such as Lorenz and Hénon-like attractors and enjoy strong statistical properties. It is natural to conjecture (or at least hope) that most dynamical systems fall into these two extreme situations. We describe a numerical test for such a conjecture/hope and apply this to the logistic map where the conjecture holds by a theorem of Lyubich, and to the 40-dimensional Lorenz-96 system where there is no rigorous theory. The numerical outcome is almost identical for both (except for the amount of data required) and provides evidence for the validity of the conjecture.
光滑动力系统中持续出现的动力学现象涵盖了从规则动力学（周期、准周期）到强混沌动力学（阿诺索夫、一致双曲、由杨塔建模的非一致双曲）。后者包括许多经典例子，如洛伦兹和类亨农吸引子，并具有很强的统计特性。很自然地会推测（或者至少希望）大多数动力系统都属于这两种极端情况。我们描述了针对这种推测/希望的一个数值检验，并将其应用于逻辑斯谛映射（根据柳比奇的一个定理，该推测在逻辑斯谛映射中成立）以及没有严格理论的40维洛伦兹 - 96系统。两者的数值结果几乎相同（除了所需的数据量），并为该推测的有效性提供了证据。
