标题： 使用动力系统理论量化渐近Lenia中的复杂性 显示英文标题

标题： Using Dynamical Systems Theory to Quantify Complexity in Asymptotic Lenia

摘要： 连续细胞自动机（CCAs）从离散查找表演变为连续偏微分方程（PDE）公式，以寻找新的复杂形式。 尽管在定性行为方面有所创新，分析方法却落后了，这强化了涌现复杂性难以简单解释的观点。 在本文中，我们证明了渐近Lenia的PDE公式可以通过动力系统理论进行严格的分析。 我们将对称性、吸引子、李雅普诺夫指数和分形维数的概念应用于数学上表征复杂行为。 我们的贡献包括：（1）对四种不同的解类（孤子、旋转器、周期性和混沌模式）的数学解释，（2）存在具有分形维数$>4$的全局吸引子的条件，（3）将Kaplan-Yorke维数识别为CCAs的有效复杂度度量，（4）一种高效的开源实现，用于计算CCAs的李雅普诺夫指数和协变李雅普诺夫向量。 最后，我们确定了在更广泛的CCAs类别中实现复杂行为的最小属性集。 这个框架为理解和测量人工生命系统中的复杂性提供了基础。 显示英文摘要

摘要： Continuous cellular automata (CCAs) have evolved from discrete lookup tables to continuous partial differential equation (PDE) formulations in the search for novel forms of complexity. Despite innovations in qualitative behavior, analytical methods have lagged behind, reinforcing the notion that emergent complexity defies simple explanation. In this paper, we demonstrate that the PDE formulation of Asymptotic Lenia enables rigorous analysis using dynamical systems theory. We apply the concepts of symmetries, attractors, Lyapunov exponents, and fractal dimensions to characterize complex behaviors mathematically. Our contributions include: (1) a mathematical explanation for the four distinct solution classes (solitons, rotators, periodic and chaotic patterns), (2) conditions for the existence of a global attractor with fractal dimension $>4$, (3) identification of Kaplan-Yorke dimension as an effective complexity measure for CCAs, and (4) an efficient open-source implementation for calculating Lyapunov exponents and the covariant Lyapunov vectors for CCAs. We conclude by identifying the minimal set of properties that enable complex behavior in a broader class of CCAs. This framework provides a foundation for understanding and measuring complexity in artificial life systems.