Convergence of trajectories to individual fixed points

Prove that every trajectory of the infinite-dimensional symmetric mean-field Oscillator Ising Machine converges to an individual fixed point, rather than merely having a connected set of fixed points as its accumulation set.

Background

The paper establishes that, in the constant-interaction case, every trajectory is precompact and that its accumulation set is a nonempty, compact, connected union of fixed points lying on one energy level. These results do not establish convergence to a single equilibrium in the infinite-dimensional limit.

The authors discuss two standard approaches—isolated fixed points and a Łojasiewicz inequality—and explain why both fail in general. They also propose reducing the problem to convergence of the finite-dimensional first order parameter z1(t)z_1(t), but note that the available energy-dissipation estimates do not provide the integrability needed to conclude convergence.

References

While Lemmas \ref{fixed_point_lemma} and \ref{stability_lemma} fully characterize the set of fixed points and their stability, and Lemma \ref{accumulation_lem} shows that the set of accumulation points of each trajectory consists of a connected union of fixed points, these results do not necessarily imply that either (1) all trajectories converge to individual fixed points, or (2) that `most'' trajectories converge to asymptotically stable fixed points. For practical purposes, we are rescued (as we explain below), but in the infinite-dimensional limit, the question is much more subtle. We provide in this setting a conditional convergence result based on the order-parameter reductionorder_param_v`, but ultimately, the questions (1) and (2) remain open regarding the infinite-dimensional limit.

— Mean-Field Oscillator Ising Machines: Gradient Flows and Classification of Limit Solutions  (2608.16025 - Venkatakrishnan et al., 17 Aug 2026) in Section 3, subsection “The Convergence Question” (Section \ref{subsec: convergence question})