Unveiling Attractor Cycles in Large Language Models: A Dynamical Systems View of Successive Paraphrasing

TL;DR

This work treats successive paraphrasing by large language models as a discrete dynamical system, modeling the process as $T_{n+1}=P(T_n)$ over a finite text space $\mathcal{T}$. It empirically uncovers robust 2-period attractor cycles across multiple models and languages, showing that generation converges to a limited, oscillatory set of paraphrases rather than exploring vast linguistic variation. The study demonstrates that increasing randomness or alternating prompts only modestly disrupts these cycles, with invertibility of paraphrase tasks further supporting attractor formation. It then proposes perturbations, history-aware transformations, and perplexity-based sampling to mitigate small cycles, and validates a data-augmentation benefit when these strategies suppress attractors, offering a dynamical-systems perspective on LLM expressivity and avenues for expanding generative diversity.

Abstract

Dynamical systems theory provides a framework for analyzing iterative processes and evolution over time. Within such systems, repetitive transformations can lead to stable configurations, known as attractors, including fixed points and limit cycles. Applying this perspective to large language models (LLMs), which iteratively map input text to output text, provides a principled approach to characterizing long-term behaviors. Successive paraphrasing serves as a compelling testbed for exploring such dynamics, as paraphrases re-express the same underlying meaning with linguistic variation. Although LLMs are expected to explore a diverse set of paraphrases in the text space, our study reveals that successive paraphrasing converges to stable periodic states, such as 2-period attractor cycles, limiting linguistic diversity. This phenomenon is attributed to the self-reinforcing nature of LLMs, as they iteratively favour and amplify certain textual forms over others. This pattern persists with increasing generation randomness or alternating prompts and LLMs. These findings underscore inherent constraints in LLM generative capability, while offering a novel dynamical systems perspective for studying their expressive potential.

Figures (15)

• Figure 1: An illustration of successive paraphrasing using GPT-4o-mini: Here, $T_0$ denotes the original human-written text, while $T_i$ indicates the i-th round of paraphrases. The nodes depicted in the lower section represent valid paraphrases for the input sentence, with distance reflecting textual variation. Successive paraphrases generated by LLMs are confined to alternating between two limited clusters, represented as blue and orange nodes.
• Figure 2: The difference confusion matrix for successive paraphrasing, where EN and ZH denotes English and Chinese sentence-level paraphrase generation accordingly. Both the x and y axes represent paraphrases at each step, and the value at the ($i$-th, $j$-th) grid position indicates the difference between the paraphrases at the $i$-th and $j$-th positions. A darker color indicates a smaller difference value between two paraphrases. The black arrow underlines the differences between $T_i$ and $T_{i-2}$, and averaging these values and subtracting the result from 1 gives our 2-period degree $\tau$.
• Figure 3: Convergence of perplexity, reverse perplexity, and generation diversity. The left and middle plots show that as the number of steps increases, both perplexity and reverse perplexity decrease steadily until they reach their lower bounds. The right plot shows that generation decreases as perplexity decreases.
• Figure 4: The difference confusion matrix for four tasks beyond paraphrasing. Note that in translations, the difference between texts in two different languages is set to one.
• Figure 5: The difference confusion matrices for model variation and prompt variation.