Lefranc Marc, Morant Pierre-Emmanuel, Nizette Michel
Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523 Centre d'Etudes et de Recherches Lasers et Applications, Université des Sciences et Technologies de Lille, Cedex, France.
Philos Trans A Math Phys Eng Sci. 2008 Feb 28;366(1865):559-67. doi: 10.1098/rsta.2007.2110.
The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.
决定论原理表明动力学状态完全决定未来的时间演化，它是非线性动力学和混沌理论的基石。由于它排除了两条状态空间轨迹相交的情况，所以它是以嵌入奇异吸引子中的不稳定周期轨道的纽结理论特征为基础的混沌拓扑分析的核心要素。然而，纽结理论仅适用于三维系统。尽管如此，决定论在任何维度都适用。我们提出了一个替代框架，在这个框架中，通过在三角剖分曲面上构建保定向动力学来强化这一原理，并且发现在三维空间中，我们的方法能数值预测马蹄映射周期轨道的正确拓扑熵。
