Generic convergence to asymptotically stable fixed points

Prove that, in the infinite-dimensional symmetric mean-field Oscillator Ising Machine, most trajectories converge to asymptotically stable fixed points, for an appropriate genericity or measure-theoretic notion of “most.”

Background

For finite-dimensional Oscillator Ising Machine dynamics, the authors explain that analyticity and stable-manifold theory imply convergence of almost every trajectory to a local minimum. They explicitly state that this conclusion is not established for the infinite-dimensional mean-field limit.

The paper notes that stable-manifold theory in infinite dimensions is delicate and technical, and that developing the required theory is outside the scope of the work. Thus, the generic convergence of trajectories to asymptotically stable equilibria remains unresolved independently of the question of convergence of every individual trajectory.

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})