Throughout the literature on Neural Cellular Automata (NCAs), it is often taken for granted that the systems learn attractors. This is shown through evolving the system for many timesteps and noting visual similarity to the goal state. There remain many questions after such an analysis. Namely, what kind of attractors do we have? Is their behavior ordered or chaotic? Can we estimate stability over very long time horizons? What really happens in the attractor when perturbations are applied? In this paper, we present a case study to help answer these questions, with methods drawn from the literature on dynamical systems theory. We use the growing gecko NCA of Mordvintsev et al. (2020) with deterministic cell updates as a case study. To the best of the authors' knowledge, we present the first visualizations of NCA attractor dynamics. We also analyze them using the Lyapunov and Fourier spectra, to reveal that the NCA displays oscillatory, periodic and quasi-periodic behavior, and that these behaviors arise early during training. This challenges the belief that NCAs learn fixed point attractors. Finally, we show that large perturbations to the attractor states can throw the NCAs into a secondary mode separate from the original attractor. We hope that this initial foray into NCA attractor dynamics expands the toolkit for NCA researchers to analyze the robustness and stability of their systems.

翻译：在神经细胞自动机（NCA）的相关文献中，通常默认其系统能学习到吸引子。这一结论通过让系统演化多个时间步并观察其与目标状态的视觉相似性得以体现。然而，此类分析后仍存诸多疑问：例如，我们拥有何种类型的吸引子？其行为是有序还是混沌？能否在极长时间尺度上估算稳定性？当施加扰动时，吸引子内部究竟会发生什么？本文以动力系统理论文献中的方法为工具，通过案例研究来解答这些问题。我们采用Mordvintsev等人（2020）提出的确定性细胞更新壁虎生长NCA作为案例。据作者所知，本文首次实现了NCA吸引子动力学的可视化。我们还利用李雅普诺夫谱和傅里叶谱对其进行分析，揭示出NCA表现出振荡、周期性和准周期性行为，且这些行为在训练早期便已出现。这挑战了NCA学习不动点吸引子的传统认知。最后，我们证明对吸引子状态的大幅度扰动可能使NCA脱离原始吸引子，进入另一种次级模式。我们希望这一对NCA吸引子动力学的初步探索，能为NCA研究者分析系统鲁棒性与稳定性扩展工具集。