Geometric Dynamics of Agentic Loops in Large Language Models

TL;DR

This paper models the iterative processes of large language models as dynamical systems, classifying their behaviors into convergence, oscillation, or exploration, and demonstrating how prompt design influences these dynamics.

Contribution

It introduces a geometric framework for analyzing LLM iterative loops as dynamical systems, enabling prediction and control of their semantic trajectories.

Findings

Iterative paraphrasing shows contractive dynamics with stable attractors.

Iterative negation results in exploratory, unbounded dynamics.

Prompt design controls the dynamical regime of LLMs.

Abstract

Iterative LLM systems(self-refinement, chain-of-thought, autonomous agents) are increasingly deployed, yet their temporal dynamics remain uncharacterized. Prior work evaluates task performance at convergence but ignores the trajectory: how does semantic content evolve across iterations? Does it stabilize, drift, or oscillate? Without answering these questions, we cannot predict system behavior, guarantee stability, or systematically design iterative architectures. We formalize agentic loops as discrete dynamical systems in semantic space. Borrowing from dynamical systems theory, we define trajectories, attractors and dynamical regimes for recursive LLM transformations, providing rigorous geometric definitions adapted to this setting. Our framework reveals that agentic loops exhibit classifiable dynamics: contractive (convergence toward stable semantic attractors), oscillatory (cycling among attractors), or exploratory (unbounded divergence). Experiments on singular loops validate the framework. Iterative paraphrasing produces contractive dynamics with measurable attractor formation and decreasing dispersion. Iterative negation produces exploratory dynamics with no stable structure. Crucially, prompt design directly controls the dynamical regime - the same model exhibits fundamentally different geometric behaviors depending solely on the transformation applied. This work establishes that iterative LLM dynamics are predictable and controllable, opening new directions for stability analysis, trajectory forecasting, and principled design of composite loops that balance convergence and exploration.