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.
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BibTeX
@misc{isik2026decisionrelated,
title = {{Decision-Related Cognitive Signatures from Fast-Slow Dynamics: A Low-Dimensional Observation-Operator Framework}},
author = {Furkan Emre Isik and Ali Demirci},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15918},
eprint = {2609.15918},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15918},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Decision-Related Cognitive Signatures from Fast-Slow Dynamics: A Low-Dimensional Observation-Operator Framework
AU - Furkan Emre Isik
AU - Ali Demirci
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15918
UR - https://arxiv.org/abs/2609.15918
AB - 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.
ER -