论文标题
通过超几何熵得出的三个不变的奇怪吸引子
Three invariants of strange attractors derived through hypergeometric entropy
论文作者
论文摘要
提供了通过三个几何和动态不变的奇怪吸引者系统的新描述。它们是相关维度（$ \ MATHCAL {D} $）和相关熵（$ \ Mathcal {k} $），两者在过去几十年中引起了人们的注意，并且是一种新的不变性，称为相关浓度（$ \ nathcal {a} $）。相关浓度定义为重建向量之间的归一化距离，通过在无限维嵌入空间上的基本概率度量进行评估。这三个不变性确定了系统的rényi-type扩展熵的缩放性能，该熵是由Kummer的汇合高几幅功能建模的，相对于量规参数（$ρ$），与重建矢量之间的距离相连。熵功能在“微观”限制$ρ\ to \ infty $中重现了$ \ Mathcal {d} $和$ \ Mathcal {k} $的已知比例行为，同时表现出新的缩放行为，以$ \ \ \ \ \ \ \ \ \ \ \ \ \ m rathcal {a} $中的其他'macrocopopic'lim lim lim $ p \ 0 $ρ\ 0 $ρ\ 0 $ρ\ 0。通过非线性回归分析同时估算这三个不变性，而无需为每个不变的单独估计。通过离散系统和连续系统中的模拟验证了所提出的方法。
A new description of strange attractor systems through three geometrical and dynamical invariants is provided. They are the correlation dimension ($\mathcal{D}$) and the correlation entropy ($\mathcal{K}$), both having attracted attention over the past decades, and a new invariant called the correlation concentration ($\mathcal{A}$) introduced in the present study. The correlation concentration is defined as the normalised mean distance between the reconstruction vectors, evaluated by the underlying probability measure on the infinite-dimensional embedding space. These three invariants determine the scaling behaviour of the system's Rényi-type extended entropy, modelled by Kummer's confluent hypergeometric function, with respect to the gauge parameter ($ρ$) coupled to the distance between the reconstruction vectors. The entropy function reproduces the known scaling behaviours of $\mathcal{D}$ and $\mathcal{K}$ in the 'microscopic' limit $ρ\to\infty$ while exhibiting a new scaling behaviour of $\mathcal{A}$ in the other, 'macroscopic' limit $ρ\to 0$. The three invariants are estimated simultaneously via nonlinear regression analysis without needing separate estimations for each invariant. The proposed method is verified through simulations in both discrete and continuous systems.
