AI 中文总结
该研究为空间进化博弈中的时空混沌构建基于基序的框架，通过布尔线性化得出局部基序解析不稳定阈值，结合德里达斜率与渐近汉明距离获四区域图谱，揭示亚临界混沌相，并建立精确基准联系损伤传播与确定性不稳定性。
AI 中文摘要
我们为空间进化博弈中的时空混沌开发了一个基于基序的框架，并用于绘制收益平面中的动态相图。通过对模仿最佳规则进行布尔线性化，我们得出了包括入侵者、合作对、条纹界面和合作核心等局部基序的解析不稳定阈值。这些阈值从有争议基序界面的收益平衡中获得，并恢复了空间进化博弈的经典入侵阈值，在此作为混沌相的边界出现。结合德里达斜率和渐近汉明距离，我们得到了一个四区域图谱：有序、瞬态混沌、持续混沌和亚临界混沌动力学。相图由密度依赖的基序选择组织：不同的初始合作者密度激活不同的不稳定机制，但一小部分基序不稳定线始终界定了不同密度下的持续混沌区域。这个图谱揭示了一个亚临界混沌相（德里达斜率$s<1$但渐近汉明距离$d_\infty>0$），其中无穷小扰动衰减而有限振幅扰动维持混沌。基于基序的框架由一个精确的基准锚定：对于均匀背景，布尔雅可比矩阵在德里达斜率和谱半径之间产生精确对应，将损伤传播与确定性不稳定性联系起来。
英文摘要
We develop a motif-based framework for spatiotemporal chaos in spatial evolutionary games and use it to map the dynamical phase diagram in the payoff plane. Using Boolean linearization of the imitate-the-best rule, we derive analytical instability thresholds for local motifs including invaders, cooperative pairs, stripe interfaces, and cooperative cores. These thresholds are obtained from payoff balance at contested motif interfaces and recover classical invasion thresholds of spatial evolutionary games, which emerge here as boundaries of the chaotic phase. Combining the Derrida slope with the asymptotic Hamming distance, we obtain a four-region cartography: ordered, transient-chaotic, sustained-chaotic, and subcritical-chaotic dynamics. The phase diagram is organized by density-dependent motif selection: different initial cooperator densities activate different instability mechanisms, yet a small set of motif-instability lines consistently bounds the sustained-chaos region across densities. This cartography reveals a subcritical chaotic phase (Derrida slope $s<1$ but asymptotic Hamming distance $d_\infty>0$), where infinitesimal perturbations decay while finite-amplitude perturbations sustain chaos. The motif-based framework is anchored by an exact benchmark: for homogeneous backgrounds, the Boolean Jacobian yields an exact correspondence between the Derrida slope and spectral radius, linking damage spreading to deterministic instability.
