Metastable state en route to traveling-wave synchronization state

The Kuramoto model with mixed signs of couplings is known to produce a traveling-wave synchronized state. Here, we consider an abrupt synchronization transition from the incoherent state to the traveling-wave state through a long-lasting metastable state with large fluctuations. Our explanation of the metastability is that the dynamic flow remains within a limited region of phase space and circulates through a few active states bounded by saddle and stable fixed points. This complex flow generates a long-lasting critical behavior, a signature of a hybrid phase transition. We show that the long-lasting period can be controlled by varying the density of inhibitory/excitatory interactions. We discuss a potential application of this transition behavior to the recovery process of human consciousness.

Figure 1

Phase diagram of synchronization transitions (STs) in the $(p,\gamma )$ plane. $p$ is the fraction of oscillators with $K_2>0$ and $\gamma$ is the half width of the uniform distribution $g\left(\omega \right)$. The phase diagram contains the IC and $\pi$ phases in (a) and (b), and the TW phase when in (b). The solid line represents a second-order transition, and both types of dashed lines represent first-order transitions, but the transition from the IC phase to the $\pi$ phase is hybrid. H represents a hysteresis zone. The symbol $•$ at ${\gamma }_h\approx 0.78$ in (a) corresponds to the hybrid critical point of the Winfree-Pazó (WP) model. The symbols $▴$ at ${\gamma }_c\approx 0.18, •$ at ${\gamma }_p\approx 0.13$, and $\blacksquare$ at ${\gamma }_t\approx 0.064$ in (b) represent critical points across which different types of phases or phase transitions emerge. The TW phase is absent when $Q>1$, or $Q<1$ and $\gamma >{\gamma }_c$.

Figure 2

Diverse types of STs at $Q=3$ and $Q=0.5$. Green triangles and red circles denote data points of $R\left(p\right)$ obtained from simulations starting from the IC and C initial states, respectively. Solid (dashed) blue curves are self-consistency solutions representing stable (unstable) states, according to the stability criterion, Eq. (5). In (a), a first-order transition and hysteresis occur between $p_c$ and $p_{c,b}$. The inset shows a closeup near $p_c$. We emphasize that the unstable line does not continuously drop to $p_c$, but instead has a finite gap of size $\gamma /|K_1|$. In (b), a hybrid phase transition (HPT) occurs with the critical exponent ${\beta }_p=2/3$ at $p_c$. A close check of the exponent value is shown in the inset. The black line guides a slope of $2/3$. (c) The TW phase emerges at ${\gamma }_c$ and exists in the range $[p_{\ell },p_u]$. When ${\gamma }_t<\gamma <{\gamma }_c, \mathrm{IC}⇝\pi \to \mathrm{TW}\to \pi$ occur with increasing $p$. (d) At $\gamma ={\gamma }_t, p_c=p_{\ell }$; thus, $\mathrm{IC}⤍\mathrm{TW}\to \pi$ occur. The part of the $\pi$ line (indicated by arrow) that is stable according to the criterion is actually metastable. (e) When $\gamma <{\gamma }_t$ ($\gamma =0.05$), $p_{\ell }<p_c<p_u$. $R$ jumps from the IC state to the TW state, and a hysteresis occurs between the IC and TW states at $[p_{\ell },p_c]$, where $\mathrm{IC}⤍\mathrm{TW}\to \pi$ occurs. Different types of arrows distinguish the types of phase transitions: continuous ($\to$), discontinuous ($⤍$), and hybrid ($⇝$).

Figure 3

Tiered ST from the IC state to TW state through the metastable $\pi$ state. $R\left(t\right)$ was obtained at various $p$ for the system size $N=25600, Q=0.5$, and $\gamma =0.064$. (a) At $p=0.42$, the TW state appears as the steady state, and the state does as metastable. The solid and dashed lines correspond to $R\left(t\right)$ and $\left|\mathrm{\Omega }\right(t\left)\right|$, respectively. Note that both the temporal and sample-to-sample (inset) fluctuations of $R$ are large during the metastable period. In (b), the velocities of $K_2$ oscillators are averaged over each specified time interval, as indicated by the corresponding colors and cluster numbers in (a). The oscillators are indexed in ascending order of the intrinsic frequencies. We find several intermediate states with different numbers of clusters composed of oscillators with similar velocities. The number of clusters increases as the stages proceed. In (a), (c), and (d), as $p$ is increased, the metastable period becomes shorter. Subsequently, the TW state is reached shortly.

Figure 4

The flow of the order parameter in the two-step synchronization transition. (a) Plot of blue dotted curve $R\left(t\right)$ vs $t$ at $p=0.418$. The time-averaged black curve $\langle R\rangle$ is obtained using a sliding 40 -s time window centered at each $t$ with a window step of 1 s. (b)–(d) The linearized flow in the $(R,\mathrm{\Omega })$ plane. Two stable points of $\pi$ and TW states are represented by red circles, and a saddle point of the TW state is shown in green. (e) An actual flow is obtained from simulations. $\mathit{x}$ represents the starting point. (f) Frequencies of the dynamic flow passing through each state in the phase space. A few states (yellow) are active throughout the flow.