Chaos in reason: How chain-of-thought LLMs can look for an answer
TL;DR
This work applies the tools of nonlinear dynamics and chaos theory to LLMs and demonstrates that LLMs exhibit hallmark signatures of chaos, including strong sensitivity to initial conditions, manifested as intermittent, jump-like divergence of nearby trajectories combined with bounded evolution.
Abstract
Large Language Models (LLMs) have achieved remarkable performance across a wide range of tasks, yet their internal dynamics remain poorly understood. In this work, we apply the tools of nonlinear dynamics and chaos theory to LLMs. By analyzing both text and hidden state trajectories, we demonstrate that LLMs exhibit hallmark signatures of chaos, including strong sensitivity to initial conditions, manifested as intermittent, jump-like divergence of nearby trajectories combined with bounded evolution, with consistent results across different distance metrics. An exact Jacobian analysis of the Transformer's sub-blocks shows that self-attention and the feed-forward network expand and propagate perturbations, while normalization and residual connections counteract this expansion and promote stability. Recurrence plots show structural similarities between LLMs and canonical chaotic systems such as the Lorenz attractor, while dimension analysis reveals fractal structures in the hidden state space, particularly pronounced in the last layers. We propose that the nonlinear coupling induced by attention mechanisms plays a key role in driving this chaotic behavior.