Detecting the temporal structure of intermittent phase locking

This study explores a method to characterize the temporal structure of intermittent phase locking in oscillatory systems. When an oscillatory system is in a weakly synchronized regime away from a synchronization threshold, it spends most of the time in parts of its phase space away from the synchronization state. Therefore characteristics of dynamics near this state (such as its stability properties and Lyapunov exponents or distributions of the durations of synchronized episodes) do not describe the system’s dynamics for most of the time. We consider an approach to characterize the system dynamics in this case by exploring the relationship between the phases on each cycle of oscillations. If some overall level of phase locking is present, one can quantify when and for how long phase locking is lost, and how the system returns back to the phase-locked state. We consider several examples to illustrate this approach: coupled skewed tent maps, the stability of which can be evaluated analytically; coupled Rössler and Lorenz oscillators, undergoing through different intermittency types on the way to phase synchronization; and a more complex example of coupled neurons. We show that the obtained measures can describe the differences in the dynamics and temporal structure of synchronization and desynchronization events for the systems with a similar overall level of phase locking and similar stability of the synchronized state.

Figure 1

Diagram of the $({\varphi }_{x,i},{\varphi }_{x,i+1})$ first-return map. The arrows indicate all possible transitions from one region to another and the expressions next to the arrows indicate the rates for these transitions.

Figure 2

An example of the first-return map for two coupled Lorenz oscillators. All four first-return plots have the same data points (gray circles), but each subplot (a–d) presents the evolution of points from one region (IV, I, III, and II, respectively). If a point evolves from one region to another region, then we represent it as $°-*$. If a point evolves within the same region, then we represent it as $°-°$. Thus each plot shows the transitions from a corresponding part of the phase space. We can compute the transition rates $r_{1,2,3,4}$ from panels (b), (d), (c), and (a), respectively.

Figure 3

We varied parameters $a$ and $\varepsilon$ to compute the transversal Lyapunov exponent ${\lambda }_{\perp }$ and rates $r_i$. The rate $r_1$ ($r_2$) and ${\lambda }_{\perp }$ have a positive (negative) relation. Rates $r_3$ and $r_4$ present more spread and more complex relationship with ${\lambda }_{\perp }$.

Figure 4

(a) Two skew tent maps for $a=0.3$ (thick solid line) and $a=0.7$ (thin solid line), respectively. (b) Transversal Lyapunov exponent ${\lambda }_{\perp }$ (dash-dotted line) and two synchronization indices $\gamma$ (thick solid line for $a=0.3$ and thin solid line for $a=0.7$). Two $\gamma$ values are almost the same. Transversal Lyapunov exponent ${\lambda }_{\perp }$ decreases as $\varepsilon$ increases and crosses 0, when $\gamma$ approaches 1.

Figure 5

Transition rates for the linearly coupled skew tent maps for $a=0.3$ (thick solid line) and $a=0.7$ (thin solid line). We used 350 000 iterations and removed the first 50 000 iterations. Differences in the rates $r_{2,3,4}$ are clearly visible.

Figure 6

Synchronized dynamics of bidirectionally coupled Rössler oscillators in dependence on the coupling strength $\varepsilon$. Two critical values for intermittencies are ${\varepsilon }_t=0.0276$ and ${\varepsilon }_c=0.0286$. (a) Rates $r_i$ and synchronization index $\gamma$. (b) Relative frequencies of the desynchronization events of different durations (durations are measured in cycles of oscillations, i.e., in the number of iterations of the ${\varphi }_{x,i}$ map, see Sec. 2). Thick solid line represents the actual relative frequencies of the desynchronization events of different durations, while thin solid line represents the relative frequencies, which would be there under the assumption of independent transition rates. The longer cycles of desynchronization are dominant relative to short cycles.

Figure 7

Synchronized dynamics of bidirectionally coupled Lorenz oscillators in dependence on the coupling strength $\varepsilon$. Two critical values for intermittencies are ${\varepsilon }_t=6.7$ and ${\varepsilon }_c=12$. (a) Rates $r_i$ and synchronization index $\gamma$. The behaviors of the rates and the durations of desynchronization events are different from coupled Rössler oscillators. (b) Relative frequencies of the desynchronization events of different durations (durations are measured in cycles of oscillations, i.e., in the number of iterations of the ${\varphi }_{x,i}$ map, see Sec. 2). The thick solid line represents the actual relative frequencies of the desynchronization events of different durations, while the thin solid line represents the relative frequencies, which would be there under the assumption of independent transitions. Unlike Fig. 6, short desynchronization events dominate the dynamics.

Figure 8

Eyelet intermittency in unidirectionally coupled Rössler oscillators with parameters ${\varepsilon }_1=0,$ ${\varepsilon }_2=\varepsilon ,$ and control parameters ${\omega }_1=0.93,\hspace{0.5em}{\omega }_2=0.95$. Two critical points are ${\varepsilon }_t=0.0345$ and ${\varepsilon }_c=0.042.$ (a) Rates $r_i$ and synchronization index $\gamma$. (b) Relative frequencies of desynchronization events of different durations. The thick solid line stands for actual duration, while the thin solid line represents the estimates from the transition rates. The intermittent behavior is different from the bidirectionally coupled Rössler oscillators.

Figure 9

Ring intermittency in coupled Rössler oscillators with parameters ${\varepsilon }_1=0,$ ${\varepsilon }_2=\varepsilon ,$ and control parameters ${\omega }_1=1.0,\hspace{0.5em}{\omega }_2=0.95$. The ring intermittency occurs between ${\varepsilon }_t=0.1097$ and ${\varepsilon }_c=0.124.$ (a) Rates $r_i$ and synchronization index $\gamma$. (b) Relative frequencies of desynchronization events of different durations. The thick solid line stands for actual duration, while the thin solid line represents the estimates from the transition rates.

Figure 10

Rates $r_i$ and synchronization index $\gamma$ for varied $g_{\text{syn},2}$ with fixed $g_{\text{syn},1}=0.1$. Although the levels of the synchronization index $\gamma$ are similar, the rates $r_2$ and $r_4$ experience more substantial variation depending on the synaptic strength $g_{\text{syn},2}$. Small noise also exerts some influence on the rates, which is not as prominent in $\gamma$. The thick solid line is the noiseless case and the thin solid line is for the noise $W$.