Continuous and discontinuous phase transitions and partial synchronization in stochastic three-state oscillators

We investigate both continuous (second-order) and discontinuous (first-order) transitions to macroscopic synchronization within a single class of discrete, stochastic (globally) phase-coupled oscillators. We provide analytical and numerical evidence that the continuity of the transition depends on the coupling coefficients and, in some nonuniform populations, on the degree of quenched disorder. Hence, in a relatively simple setting this class of models exhibits the qualitative behaviors characteristic of a variety of considerably more complicated models. In addition, we study the microscopic basis of synchronization above threshold and detail the counterintuitive subtleties relating measurements of time-averaged frequencies and mean-field oscillations. Most notably, we observe a state of suprathreshold partial synchronization in which time-averaged frequency measurements from individual oscillators do not correspond to the frequency of macroscopic oscillations observed in the population.

Figure 1

Three-state unit with transition rates $g$.

Figure 2

(Color online) In a single population of globally coupled oscillators, two physically distinct Hopf transitions can be observed depending on the choices of $U$, $V$, and $W$. The top panel shows the synchronization order parameter $r$ as a function of the coupling strength $a$ for $(U,V,W)=(1,-2,0)$ and clearly shows characteristics of a discontinuous transition, including hysteresis. Squares represent solutions starting from ordered (mostly synchronized) initial conditions, while circles represent solutions starting from disordered (random) initial conditions. The bottom panel shows the order parameter as a function of coupling strength for $(U,V,W)=(2,-1,0)$ and displays a continuous transition with critical exponent $\beta$ given by the classical value 1/2. The inset shows a log-log plot near the critical point. For comparison, a dashed line with a slope of 1/2 is shown along with the order parameter curve (solid line) to verify this scaling relation.

Figure 3

The first Lyapunov coefficient $l_1$ as a function of the relative width parameter $\mu$ is shown for Hopf bifurcations taking place at ${ϵ}^H=\mathbf{\left(}{a}_c\right(\mu ),\mu \mathbf{)}$. The bifurcation can be either subcritical or supercritical depending on the relative width variable.

Figure 4

(Color online) Order parameter $r$ as a function of the coupling strength $a$. A subcritical Hopf bifurcation occurs for $\mu =3∕4$. Squares represent solutions starting from ordered (mostly synchronized) initial conditions, while circles represent solutions starting from disordered (random) initial conditions. Dark (blue) points correspond to population 1, ${\gamma }_1=2.5$, and light (pink) points to population 2, ${\gamma }_2=5.5$. The transition is clearly discontinuous as $a$ crosses $a_c\approx 3.55$. In addition, a region of multistability and corresponding hysteresis exists just below threshold.

Figure 5

(Color online) Order parameter $r$ vs coupling strength $a$. A supercritical Hopf bifurcation occurs for $\mu =7∕4$. Squares represent solutions starting from an ordered (mostly synchronized) initial condition, while circles represent solutions starting from a disordered (random) initial condition (repetition with other ordered and disordered initial conditions leads to essentially the same results). Dark (blue) points correspond to population 1, ${\gamma }_1=0.25$, and light (pink) points to population 2, ${\gamma }_2=3.75$. The transition is clearly continuous as $a$ crosses $a_c\approx 5.44$, and there is a noticeable absence of hysteresis. Dotted line is drawn to guide the eye. The inset indicates that, in the neighborhood of the critical point, the order parameter follows power law behavior with the correct mean-field critical exponent $(\beta =1∕2)$.

Figure 6

(Color online) Each plot shows a histogram of time-averaged frequencies (in the steady state), where the vertical axis represents the number of units (out of $N=3500$ total units) having the frequency $\overline{\omega }$. The power spectrum of $P_{1,{\gamma }_1}$ overlays each histogram. The top panel is below synchronization threshold $(a=2.65)$, while the middle $(a=3.05)$ and lower panels $(a=3.45)$ are both above threshold.

Figure 7

(Color online) Each plot shows a histogram of time-averaged frequencies (in the steady state), where the vertical axis represents the number of units (out of $N=3500$ total units) having the frequency $\overline{\omega }$. Population 1, characterized by ${\gamma }_1=2.5$, is represented by the dark (blue) histogram, while population 2, characterized by ${\gamma }_2=5.5$, is represented by the light (pink) one. Power spectra of $P_{1,{\gamma }_1}$ (dark, blue) and $P_{1,{\gamma }_2}$ (light, pink) overlay the histograms. The top panel is below synchronization threshold $(a=3.20)$, while the middle $(a=3.60)$ and lower panels $(a=3.86)$ are just above threshold. For aesthetic purposes, the horizontal range is relatively shifted in the three panels.

Figure 8

(Color online) The central figure shows a histogram of waiting times $T_{1step}$ for a sampling of $N=60$ units once the steady state has been reached. The vertical black line indicates the waiting time corresponding to the peak in the power spectrum of $P_1\left(t\right)$ (that is, the waiting time corresponding to the frequency of the macroscopic oscillations). The inset shows a similar histogram for a single unit. $a=3.05$ (above threshold), $\gamma =1$ in all plots.

Figure 9

(Color online) The central figure shows a histogram of waiting times $T_{1step}$ for a sampling of $N=30$ units from each population once the steady state has been reached. Dark (blue) represents population 1, while light (red) represents population 2. The vertical black line indicates the waiting time corresponding to the peak in the power spectrum of $P_1\left(t\right)$ (that is, the waiting time corresponding to the frequency of the macroscopic oscillations). The insets show similar histograms for single units; the top histogram is for a unit from population one, the bottom from population two. $a=3.60$ (above threshold), ${\gamma }_1=2.5$, and ${\gamma }_2=5.5$ in all plots.

Figure 10

(Color online) The top panels show the evolution of a representative sub-system ($N=30$ units of each population). Dark, medium, and light represent states 1, 2, and 3, respectively. The bottom panels show the macroscopic variable $P_{1,{\gamma }_i}$ for each population. The left panels correspond to population 1 $({\gamma }_1=2.5)$, while the right panels correspond to population 2 $({\gamma }_2=5.5)$. $a=3.60$ (above threshold) for all plots.