Unsteady phase waves in the 1D swarmalator model with inertia

Abstract: We study a one-dimensional swarmalator model with inertia. Previous studies have focused almost exclusively on the overdamped limit. We find inertia introduces a new unsteady collective state in which the rainbow order parameters undergo multiharmonic oscillations. This "thrashing" phase wave bifurcates from the model's static phase wave state through a subcritical Hopf bifurcation that coincides with a saddle-node of limit cycles. The wave itself exists in clockwise and counterclockwise symmetric pairs. For small populations we observe attractor switching between these chiral states, while for larger systems the dynamics settle onto a single branch.

• The paper shows that adding inertia to both spatial and phase dynamics preserves the async threshold K=0 and sync threshold K=J, while changing transients and destabilizing static phase waves.
• The paper derives a closed-form subcritical Hopf boundary, m_H(K)=4K/(8J²−9K²), with m=0.5 and J=1 predicting K_H≈0.598 in agreement with simulations.
• The paper identifies a thrashing phase wave with sustained multiharmonic order-parameter oscillations, dominant frequency near 0.20 in a representative case, and finite-size switching between opposite chiralities.

Overview

Swarmalators—phase oscillators that simultaneously move in space and synchronize their internal phases—have been studied almost exclusively under the overdamped assumption. This paper by O'Keeffe (2603.12531) removes that assumption by adding an inertial mass $m$ to both the spatial and phase dynamics of the one-dimensional swarmalator model, a setting chosen for its exact tractability. The central finding is that inertia leaves the stability thresholds of the async and sync states unchanged, but destabilizes the static phase wave, producing a new unsteady collective state—the "thrashing" phase wave—in which the rainbow order parameters undergo sustained multiharmonic oscillations. Since all attractors of the non-inertial 1D model are static, inertia's principal dynamical effect is to generate unsteadiness.

Model

The model is the 1D swarmalator system with second-order (inertial) dynamics on both position $x_i \in S^1$ and phase $\theta_i \in S^1$, with mean-field coupling between the two. In sum/difference coordinates $\xi_i = x_i + \theta_i$ and $\eta_i = x_i - \theta_i$, the equations take a symmetric form governed by two rainbow order parameters $U = re^{i\phi} = \langle e^{i\xi}\rangle$ and $V = se^{i\psi} = \langle e^{i\eta}\rangle$. For identical natural frequencies (set to zero), the governing equations reduce to damped-driven forms in which each coordinate feels restoring forces proportional to $Kr\sin\xi_i$, $Js\sin\eta_i$ and cross-terms.

Numerical integration reveals four collective states: async ($(r,s)=(0,0)$), sync ($x_i \in S^1$0), static phase wave (one order parameter at unity, the other at zero, realized in symmetry-related $x_i \in S^1$1-wave and $x_i \in S^1$2-wave branches), and the novel thrashing phase wave, in which the system remains phase-wave-like on average but the order parameters oscillate periodically—a state with no analogue at $x_i \in S^1$3.

Stability of sync and async: inertia-independent thresholds

Linearizing about the synchronized fixed point and separating mean modes from shape modes yields eigenvalues $x_i \in S^1$4 (rotational neutrality) and $x_i \in S^1$5 in the mean sector, while the decoupled $x_i \in S^1$6 sectors give stable eigenvalues provided $x_i \in S^1$7. The sync threshold is therefore $x_i \in S^1$8, identical to the overdamped result and independent of $x_i \in S^1$9; inertia only modifies transients, producing oscillatory ringing for large $\theta_i \in S^1$0 before settling.

For the async state, a kinetic-theory treatment of the phase-space density $\theta_i \in S^1$1 with normal-mode perturbations gives the characteristic equation $\theta_i \in S^1$2, so async is linearly stable iff $\theta_i \in S^1$3. Again the critical coupling $\theta_i \in S^1$4 is independent of both $\theta_i \in S^1$5 and the cross-coupling $\theta_i \in S^1$6, which drops out upon projection onto first Fourier harmonics. These results establish that inertia does not shift the boundaries of the ordered/disordered regimes; its effect is confined to the phase-wave branch.

Hopf instability of the phase wave

Perturbing the $\theta_i \in S^1$7-wave ($\theta_i \in S^1$8, $\theta_i \in S^1$9) with first-harmonic ansätze in $\xi_i = x_i + \theta_i$0 leads to a quartic characteristic equation reducible via $\xi_i = x_i + \theta_i$1 to a quadratic. Setting $\xi_i = x_i + \theta_i$2 yields the Hopf frequency $\xi_i = x_i + \theta_i$3 and the analytic bifurcation boundary

$\xi_i = x_i + \theta_i$4

or equivalently, for fixed $\xi_i = x_i + \theta_i$5,

$\xi_i = x_i + \theta_i$6

At $\xi_i = x_i + \theta_i$7, $\xi_i = x_i + \theta_i$8, this predicts $\xi_i = x_i + \theta_i$9, matching the numerically observed transition. The agreement between the closed-form prediction and simulation is a strong quantitative result, and the full $\eta_i = x_i - \theta_i$0 state diagram is summarized analytically.

Subcritical character and the thrashing state

The Hopf bifurcation is subcritical: the first Lyapunov coefficient $\eta_i = x_i - \theta_i$1 is positive for all tested sizes $\eta_i = x_i - \theta_i$2, and sweeping $\eta_i = x_i - \theta_i$3 through the Hopf point produces discontinuous jumps in both oscillation amplitude and frequency—ruling out supercritical scaling—and consistent with a nearby saddle–node of limit cycles. Numerically, $\eta_i = x_i - \theta_i$4 appears to coincide with $\eta_i = x_i - \theta_i$5, though no bistability between the static and thrashing waves was found despite an explicit search. The coincidence of the Hopf point with the saddle–node of cycles is asserted from numerics rather than proven analytically, which remains an open point.

The thrashing phase wave itself exhibits a sharp spectral peak at dominant frequency $\eta_i = x_i - \theta_i$6 (period $\eta_i = x_i - \theta_i$7) at $\eta_i = x_i - \theta_i$8, $\eta_i = x_i - \theta_i$9, indicating a well-defined collective oscillation in which particles drift slowly while relative phase relations oscillate. A further finite-size phenomenon is attractor switching: for small $U = re^{i\phi} = \langle e^{i\xi}\rangle$0, the system intermittently alternates between the clockwise ($U = re^{i\phi} = \langle e^{i\xi}\rangle$1) and counterclockwise ($U = re^{i\phi} = \langle e^{i\xi}\rangle$2) chiral phase waves; switching becomes rarer with increasing $U = re^{i\phi} = \langle e^{i\xi}\rangle$3 and vanishes in the large-$U = re^{i\phi} = \langle e^{i\xi}\rangle$4 limit, where a single chirality is locked in. This switching was verified across three independent integrators (RK45, DOP853, Radau), confirming it is dynamical rather than numerical.

Limitations and open questions

Several caveats qualify these results. The analysis assumes identical natural frequencies; distributed frequencies, disorder in coupling, delay, and noise are explicitly deferred to future work, so the robustness of the subcritical Hopf scenario to heterogeneity is unknown. The identification $U = re^{i\phi} = \langle e^{i\xi}\rangle$5 rests on numerical evidence, and the absence of detected bistability near a putative saddle–node of limit cycles is not fully explained. Attractor switching is observed only at small $U = re^{i\phi} = \langle e^{i\xi}\rangle$6, and its mechanism—presumably noise-induced hopping between symmetry-related attractors—is not characterized quantitatively. Finally, whether analogous inertial unsteadiness arises in higher-dimensional swarmalator models remains open.

Conclusion

Adding inertia to the tractable 1D swarmalator model preserves the async and sync stability boundaries exactly but destabilizes the static phase wave through a subcritical Hopf bifurcation whose boundary is obtained in closed form and matches simulations. The resulting thrashing phase wave—with multiharmonic order-parameter oscillations and small-system chiral switching—demonstrates that the interplay of synchronization and self-assembly permits inertia-induced unsteadiness absent in identical-frequency Kuramoto oscillators, where only static outcomes occur.