Understanding the Effects of Hyper-Connections on Self-Attention Dynamics: A Bifurcation Analysis

A central challenge in understanding deep sequence models is characterizing how token representations evolve across layers: whether they collapse, diverge, or converge to nontrivial structures. We study this question for Transformers with hyper-connections, whose learnable cross-layer coupling makes the stability of representation collapse analytically tractable. In contrast to standard residual connections, hyper-connections introduce additional parameters that directly control the hyperbolicity of the zero fixed point, at which the local stability is fully determined by the Jacobian eigenvalues, thereby enabling a rigorous and precise bifurcation analysis of the dynamical system underlying the model. In particular, we derive a continuous-depth ODE limit of self-attention with hyper-connections and use bifurcation theory to characterize its token dynamics. For the 1-stream case, we obtain a closed-form critical threshold at which a pitchfork bifurcation occurs, separating representation collapse from divergence or convergence to nontrivial stable states. For the 2-stream case, the richer coupling structure yields two distinct stability boundaries, corresponding to qualitatively different instabilities: one arising when a real eigenvalue crosses zero, altering the number of nearby equilibria (static type), and another when a complex-conjugate pair reaches the imaginary axis, leading to oscillatory behavior (Hopf type). These phenomena have no counterpart in the 1-stream setting. Experiments on pretrained hyper-connected language models validate our theory and show that unstable regimes yield stronger performance.