Phase-shift inversion in oscillator systems with periodically switching couplings

A system's response to external periodic changes can provide crucial information about its dynamical properties. We investigate the synchronization transition, an archetypical example of a dynamic phase transition, in the framework of such a temporal response. The Kuramoto model under periodically switching interactions has the same type of phase transition as the original mean-field model. Furthermore, we see that the signature of the synchronization transition appears in the relative delay of the order parameter with respect to the phase of oscillating interactions as well. Specifically, the phase shift becomes significantly larger as the system gets closer to the phase transition, so that the order parameter at the minimum interaction density can even be larger than that at the maximum interaction density, counterintuitively. We argue that this phase-shift inversion is caused by the diverging relaxation time, in a similar way to the resonance near the critical point in the kinetic Ising model. Our result, based on exhaustive simulations on globally coupled systems as well as scale-free networks, shows that an oscillator system's phase transition can be manifested in the temporal response to the topological dynamics of the underlying connection structure.

Figure 1

A typical behavior of the order parameter $\Delta \left(t\right)$ and the interaction density $\rho \left(t\right)$. The system size $N=800$, the edge frequency $\Omega =1.00$, and the time series of $\Delta \left(t\right)$ depending on $K$ are shown as smooth curves in (a) and $\rho \left(t\right)$ (independent of $K$) as black triangle waves in (a). The $\Delta$–$\rho$ diagram for $t>20$ after the transient behavior is shown in (b). All the curves are results averaged over 50 samples.

Figure 2

FSS scaling collapse of the temporally averaged order parameter $\overline{\Delta }$ for various system sizes with the critical exponents of the original MF Kuramoto model [$K_c=1.62\left(1\right)$, $\beta =1/2$, and $\overline{\nu }=5/2$]. The edge frequency is $\Omega =1.00$.

Figure 3

The difference $A$ between maximum and minimum values of $\Delta \left(t\right)$ oscillation and the area $S$ enclosed by the loop in $\Delta$–$\rho$ the diagram are shown in the $K$–$\Omega$ plane. The system size $N=800$, and the color-coded values are $A$ (a), $A\Omega$ (b), $S$ (c), and $S\Omega$ (d). The time series $\Delta \left(t\right)$ for $t>20$ after the transient behavior is used, and all the results are averaged over 50 samples.

Figure 4

$A\Omega$ (a) and $S\Omega$ (b) as functions of $K$, for various $\Omega$ values. The black vertical lines correspond to the MF critical coupling strength $K_c^{\mathit{MF}}=2\sqrt{2/\pi }$.

Figure 5

Height difference between $\Delta \left({\rho }_{\max }\right)$ and $\Delta \left({\rho }_{\min }\right)$ for the globally coupled case (a) and SFNs with $\gamma =6.50$ (b), $3.75$ (c), and $2.20$ (d). Note that the rescaled coupling strength $KN/\langle k\rangle$, where $\langle k\rangle$ is set as 20, is used for (b)–(d), to compare with the globally coupled case (a) with the same Eq. (1). The same system size $N=800$ is used for all the cases.