Graph-Induced Rotational Twisted States in Systems of Identical Oscillators
arXiv:2607.11833
The paper investigates a new type of collective motion called rotational twisted states in identical Stuart‑Landau oscillators coupled on graphs, showing how graph asymmetry can induce rotating patterns that coexist with synchronization.
Abstract
In this article, we study a new class of collective motion for identical coupled Stuart-Landau oscillators on graphs. This model was previously known to converge to the synchronized state for a certain class of initial data. Here, we show that when the interaction matrix is circulant, there exists another class of attractors, in which the particles are uniformly distributed on a circle, reminiscent of the \textit{twisted states} known for the Kuramoto model, but which rotate around the origin. However, contrary to the classical Kuramoto case, here the rotation is caused by the asymmetry of the graph structure, and not by the natural frequencies. We identify conditions for the existence and local stability of the \textit{rotational twisted states}, both in the Stuart-Landau model and in the Kuramoto model. We also provide sufficient conditions for the existence and stability of the synchronized state, which can co-exist with the rotational twisted state in a certain parameter region. We provide a generalization of the class of interaction matrices able to generate rotational twisted states via leader-follower interaction matrices. We show that heterogeneous rotational twisted states can also exist for a system of heterogeneous oscillators, attracted to different individual target amplitudes. This study is accompanied by numerical simulations that illustrate the possible behaviors of the system, which also include metastable dynamics and a chimera state.
39 pages, 8 figures