AI 中文总结
该研究提出一个二维快慢动力系统框架，通过观测算子将轨迹投影为二元信号，以生成重复决策中的时间结构，并测试其生成充分性而非验证认知机制。
AI 中文摘要
重复决策表现出时间结构，如持续性、方向依赖的切换、循环交替和突变转换。我们考察了一个二维快慢动力系统的生成充分性。该系统将三次快方程与线性慢反馈相结合，并通过其平衡几何、迹-行列式结构、平衡折叠轨迹、候选霍普夫边界和奇异临界流形进行分析。一个显式观测算子将连续轨迹投影为标量信号，并应用二元读出，将潜在状态空间动力学与可观测行为分离。分析确立了一个唯一平衡区域，并在适当参数下，确立了一个具有中心鞍点的三平衡楔形。外部平衡仅在迹为负时具有吸引性。分析还识别了普通鞍结所需的简单零条件、与振荡不稳定性相容的迹为零且行列式为正的谱边界，以及临界流形的吸引和排斥分支。预设的非自治扫描数值上展示了方向依赖的切换（T1）、瞬态情景性循环切换（T2）和突发的局部状态转变（T3）。与长时间状态保持相关的吸引平衡区域（T4）在解析上被刻画。所得对应关系旨在作为可观测时间组织层面的生成充分性测试，而非对潜在认知机制的识别或经验验证。
英文摘要
Repeated decisions exhibit temporal structures such as persistence, direction-dependent switching, recurrent alternation, and abrupt transitions. We examine the generative sufficiency of a two-dimensional fast-slow dynamical system. The system combines a cubic fast equation with linear slow feedback and is analyzed through its equilibrium geometry, trace-determinant structure, equilibrium-fold loci, candidate Hopf boundaries, and singular critical manifold. An explicit observation operator projects continuous trajectories to a scalar signal and applies a binary readout, separating latent state-space dynamics from observable behavior. The analysis establishes a unique-equilibrium regime and, for suitable parameters, a three-equilibrium wedge with a central saddle. The outer equilibria are attracting only where their traces are negative. The analysis also identifies the simple-zero condition required for ordinary saddle-nodes, trace-zero positive-determinant spectral boundaries compatible with oscillatory instability, and the attracting and repelling branches of the critical manifold. Prescribed nonautonomous sweeps numerically illustrate direction-dependent switching (T1), transient episodic recurrent switching (T2), and an abrupt localized regime shift (T3). The attracting-equilibrium regime associated with prolonged state retention (T4) is characterized analytically. The resulting correspondence is intended as a test of generative sufficiency at the level of observable temporal organization, rather than as an identification or empirical validation of latent cognitive mechanisms.
Comments28 pages, 6 figures
