Stability of Phase-Locked States in Signed Kuramoto Networks: Structure versus Adaptation

Abstract

Adaptive Kuramoto models admit a variety of nontrivial phase-locked configurations, including antipodal and rotating-wave states. A central open question is whether the observed persistence of such configurations can be attributed to intrinsic properties of the associated signed interaction networks, or whether it relies essentially on adaptive coupling dynamics. To address this question, we study the stability of antipodal and rotating-wave phase configurations on fixed signed networks that preserve the same phase symmetries but are not generated by adaptive dynamics. We show that for two canonical classes of static signed networks, stability is highly constrained, with unstable modes persisting under parameter variations generically, and we characterize how adaptive coupling influences invariant sets and basins of attraction for the configurations where stability is permitted. Taken together, these results show that while static network structure imposes severe constraints on the stability of phase-locked configurations, adaptive coupling dynamics organize and delineate their robustness when stability is permitted.

Figures (8)

• Figure 1: Graph structures $K=(\kappa_{ij})$ when equilibrium are in the form of (a) complete synchronization and (b) antipodal states.
• Figure 2: Color-coded adjacency matrix of network $\kappa_{ij}=-\sin(\theta_i-\theta_j+\beta)$
• Figure 3: Local stability of the complete synchronization state based on \ref{['prop_eigval_com_syn']}. The region where the stability cannot be determined just by $a$ and $b$, which was left blank in (a), is depicted in (b) based on $b/a$ and the maximal group proportion.
• Figure 4: The admissible range of $p$ to satisfy \ref{['lam_cond']}. Blue dots and yellow crosses indicate the upper bound and lower bounds, respectively, and the yellow line represents the admissible set of $p$. The absence of markers implies that the admissible set is empty. As demonstrated, the admissible range of $W$ decreases with increasing $m$, suggesting a roughly inverse relationship between the two parameters. All simulations are based on $N=100$.
• Figure 5: Schematic illustration of the invariant set $\mathcal{A}_{c,\delta}$. (Left) Phase configuration associated with a partition $\{\mathcal{N}_1,\mathcal{N}_2\}$, where oscillators in $\mathcal{N}_1$ (resp. $\mathcal{N}_2$) have phases confined to $[0,c]$ (resp. $[\pi,\pi+c]$). (Right) Corresponding coupling matrix $K$, where intra-group couplings satisfy $\kappa_{ij}\in[\delta,1]$ and inter-group couplings satisfy $\kappa_{ij}\in[-1,-\delta]$.

Theorems & Definitions (10)

• Proposition 2.1: Phase-locked solutions
• Proposition 3.1: Local Stability of Complete Synchronization
• Proposition 3.2: Local Stability of Antipodal States
• Lemma 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Remark 4.4