Geometric Dynamics of Agentic Loops in Large Language Models

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.