利用动力系统理论量化渐近Lenia中的复杂性
Using Dynamical Systems Theory to Quantify Complexity in Asymptotic Lenia
AI总结：
本文利用动力系统理论对渐近Lenia的PDE形式进行严格分析，通过对称性、吸引子、李雅普诺夫指数和分形维数等概念，解释了四种解类，提出了全局吸引子条件，并确定Kaplan-Yorke维数为有效复杂性度量，为人工生命系统的复杂性量化奠定基础。
摘要：
连续元胞自动机（CCAs）已从离散查找表演变为连续偏微分方程（PDE）形式，以探索新颖的复杂性形式。尽管在定性行为方面有所创新，但分析方法却滞后，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形式能够利用动力系统理论进行严格分析。我们应用对称性、吸引子、李雅普诺夫指数和分形维数的概念来数学地表征复杂行为。我们的贡献包括：（1）对四种不同解类（孤子、旋转子、周期和混沌模式）的数学解释；（2）具有分形维数$>4$的全局吸引子存在的条件；（3）确定Kaplan-Yorke维数作为CCAs的有效复杂性度量；（4）一个用于计算CCAs的李雅普诺夫指数和协变李雅普诺夫向量的高效开源实现。最后，我们确定了在更广泛的CCAs类别中实现复杂行为所需的最小属性集。该框架为理解和测量人工生命系统中的复杂性提供了基础。
英文摘要：
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.
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