Decision-Related Cognitive Signatures from Fast-Slow Dynamics: A Low-Dimensional Observation-Operator Framework

Abstract

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.