Synchronization of coupled Boolean phase oscillators

We design, characterize, and couple Boolean phase oscillators that include state-dependent feedback delay. The state-dependent delay allows us to realize an adjustable coupling strength, even though only Boolean signals are exchanged. Specifically, increasing the coupling strength via the range of state-dependent delay leads to larger locking ranges in uni- and bidirectional coupling of oscillators in both experiment and numerical simulation with a piecewise switching model. In the unidirectional coupling scheme, we unveil asymmetric triangular-shaped locking regions (Arnold tongues) that appear at multiples of the natural frequency of the oscillators. This extends observations of a single locking region reported in previous studies. In the bidirectional coupling scheme, we map out a symmetric locking region in the parameter space of frequency detuning and coupling strength. Because of the large scalability of our setup, our observations constitute a first step towards realizing large-scale networks of coupled oscillators to address fundamental questions on the dynamical properties of networks in a new experimental setting.

Figure 1

Graphical description of the three principal functional blocks comprising a phase-locked loop (PLL).

Figure 2

(a) Symbol of an xor logic gate that is used as a one-bit phase detector and its associated look-up table. The low ($V_L$) and high ($V_H$) Boolean voltages are denoted by symbols 0 and 1, respectively. (b) The error signal $V_0\oplus V_1$ resulting from the phase detector with two phase-shifted input signals $V_0$ and $V_1$ of frequency $f_0=f_1=30.0\pm 0.1\text{MHz}$. The phase difference is highlighted by shaded regions at a falling and a rising Boolean transition. The wave form is obtained from an experimental implementation on a field-programmable gate array, specifically the model Altera Cyclone IV EP4CE115F29C7, which is used for all experimental implementations in this paper.

Figure 3

(a) Construction of delay line ${\tau }_n$ with a series of $n$ cascaded buffers with individual propagation time ${\tau }_{\mathrm{LG}}$. (b) Schematic of the CO block of our BPO. (c) Locking of the CO block to an externally generated control signal $V_c$ with frequency $f=60.0\pm 0.1\mathrm{MHz}$. Also shown are the output wave form $V$ and the signals from the delay lines that are input to the switch $V_{m0}$ and $V_{m1}$. The parameters of the CO block are $n=65$ and $k=10$. The gate propagation delay as measured for buffer gates is ${\tau }_{\text{LG}}=0.275\pm 0.010\mathrm{ns}.$

Figure 4

(a) Experimental setup of the master-slave coupling scheme for two BPOs. (b) Construction of the slave BPO comprising the CO block labeled $(n,k)$ (see Fig. 3) and an xor-based phase detector (PD) block. (c) Wave forms of the master oscillator (generated externally with wave form generator Tektronix AFG3251) $V_m$ of frequency $f_m=28.6\pm 0.1\mathrm{MHz}$, the output wave form of the slave oscillator $V_s$, and the error signal $V_{c,ms}=V_m\oplus V_s$. The parameters of the CO block in the BPO are the same as the ones in Fig. 3.

Figure 5

(a) Experimental devil's staircase for a BPO with $k=10$ showing synchronization regions as horizontal lines. For the three integer synchronization regions $f_m:f_s=1:1$, $2:1$, and $3:1$, the theoretical limits are shown with red dashed lines. (b) Experimental Arnold tongues for the three integer synchronization regions as a function of $k$ and the theoretical limits of the regions marked with red dashed lines. Other Arnold tongues are not shown. Setup and experimental parameters are as in Fig. 3.

Figure 6

(a) Block representation of the mathematical model for the master-slave setup. LPF${}_{x,y}$ are low-pass filters with time constants ${\tau }_x$ and ${\tau }_y$, respectively. (b) Numerical simulation with parameters as in Fig. 3. The variables $y_m$, $y_s$, and $y_{c,ms}$ are the model representations of the experimental voltages $V_m$, $V_s$, and $V_{c,ms}$ in Fig. 4. The parameters are the same as in the experiment; see Fig. 3.

Figure 7

(a) Devil's staircase and (b) Arnold tongues from numerical simulations, similar to Fig. 5. The parameters are the same as in Fig. 3.

Figure 8

(a) Setup of the two mutually coupled oscillators. The coupling is realized with two directed links of equal strength $k$. (b) Experimental and (c) numerical synchronization plane in parameter space $(\Delta n,k)$ measuring the absolute value of the normalized beat frequency $f_b$ for $\Delta n\in \left\{-15,\cdots ,15\right\}$, $k\in \left\{0,\cdots ,15\right\}$, and $n_1=65$ ($n_2=n_1+\Delta n$). The frequency is color coded; white color indicates normalized beating frequencies $f_b<0.0025$. The red dashed lines indicate the analytic synchronization boundary $k=\left|n_2-n_1\right|$. The parameters are the same as in Fig. 3.