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.

Figures (8)

• Figure 1: Collective states of the inertial 1D swarmalator model, Eqs. (1)--(2).. Top row: snapshots of swarmalators in the $(\xi_i,\eta_i)$ plane. Bottom row: time series of the rainbow order parameters $r=|\langle e^{i\xi}\rangle|$, $s=|\langle e^{i\eta}\rangle|$, and the normalized mean speed $v/v_{\max}$. From left to right: async, sync, phase wave, and thrashing phase wave. Parameter values $(K,m)$ are indicated in each panel. Simulations used $N=200$, $dt=0.05$, and $m=0.5$.
• Figure 2: Order parameters $r$, $s$, and $v/v_{\max}$ as a function of coupling $K$, for $m=0.5$, $J=1$, $N=500$. Each point is averaged over 5 random initial conditions. Simulations used RK45 with $dt=0.05$, total time $T=2000$, and transient $T_{\rm trans}=1000$ discarded.
• Figure 3: Effect of inertia on the transient dynamics of $r(t)$, where we set $r(t)>s(t)$ without loss of generality, starting from random initial conditions. Top: sync state ($K=2$); bottom: async state ($K=-1.5$, starting near $r=1$). Curves show $m=0,1,2,4,8$. Larger $m$ slows the approach to steady state and introduces oscillatory ringing before settling. Simulations used $N=500$, $dt=0.05$.
• Figure 4: Evidence for subcritical Hopf bifurcation at $m=0.5$, $N=100$. (a) Oscillation amplitude of $s(t)$ vs $K$ from random initial conditions, showing a finite jump at $K_c \approx 0.598$ rather than the continuous $\sqrt{K_H - K}$ scaling expected for a supercritical Hopf. (b) Frequency of oscillation vs $K$, showing a discontinuous jump at $K_H$.
• Figure 5: Characterization of the thrashing phase wave at $K=0.3$, $m=0.5$. (a) Time series of $r(t)$ and $s(t)$. (b) Power spectrum of $s(t)$, with dominant frequency $f^* \approx 0.20$. (c) Trajectories $\xi_i(t)$ (solid) and $\eta_i(t)$ (dashed) for a typical swarmalator. Simulations used $N=500$, $dt=0.05$, with the first $100$ time units discarded as transient.