Experimental confirmation of chaotic phase synchronization in coupled time-delayed electronic circuits

We report the experimental demonstration of chaotic phase synchronization (CPS) in unidirectionally coupled time-delay systems using electronic circuits. We have also implemented experimentally an efficient methodology for characterizing CPS, namely, the localized sets. Snapshots of the evolution of coupled systems and the sets as observed from the oscilloscope confirming CPS are shown experimentally. Numerical results from different approaches, namely, phase differences, localized sets, changes in the largest Lyapunov exponents, and the correlation of probability of recurrence $\left(C_{CPR}\right)$ corroborate the experimental observations.

Figure 1

Schematic diagram of the unidirectionally coupled time-delay analog circuits with threshold nonlinearity. ND1 and ND2 are nonlinear device units; delay unit consists of ten pairs of capacitors of 470 nF and inductors of 12 mH. $R_1=R_2=1.86\text{k}\Omega$, $C_1=C_2=100\text{nF}$, and $A$ is the op-amp difference amplifier.

Figure 2

ND unit of Fig. 1: actual circuit implementation of the nonlinear activation function consisting of diodes $D_1=D_2=1N4148$, resistor $R_1=R_7=1\text{k}\Omega$, $R_2=R_3=10\text{k}\Omega$, $R_4=2\text{k}\Omega$, $R_5=3\text{k}\Omega$, $R_6=10.4\text{k}\Omega$, and $R_8=5\text{k}\Omega$, and threshold control voltages $V_1=V_2=0.7\text{V}$ along with different amplifying stages $\left(O{A}_1=O{A}_2=O{A}_3=uA741\right)$.

Figure 3

(Color online) Snapshots of the time evolution of both coupled systems indicating the existence of CPS in (a) chaotic regime and (b) hyperchaotic regime. $x$ axis: time $\left(1\text{unit}=1.0\text{ms}\right)$; $y$ axis: voltage $\left(1\text{unit}=1.0\text{V}\right)$.

Figure 4

(Color online) Phase difference for different values of $\epsilon =0.3$, 0.5, 0.7, and 0.9 and $\tau =1.33$.

Figure 5

(Color online) Experimental characterization of CPS using the framework of localized sets in the chaotic regime (i) for $\tau =1.33$ and in the hyperchaotic regime (ii) for $\tau =6.0$. Sets in the drive and the response systems are distributed in (a) and (b) for $\epsilon =0.3$ indicating the asynchronous state and localized in (c) and (d) for $\epsilon =0.9$ indicating CPS, respectively. $x$ axis: voltage $U\left(t\right)$ $\left(1\text{unit}=0.5\text{V}\right)$; $y$ axis: voltage $U\left(t-T_d\right)$ $\left(1\text{unit}=2.0\text{V}\right)$.

Figure 6

(Color online) Numerical confirmation of CPS using the framework of localized sets in the chaotic regime (i) for $\tau =1.33$ and in the hyperchaotic regime (ii) for $\tau =6.0$. Sets in the drive and the response systems are distributed in (a) and (b) for $\epsilon =0.3$ indicating the asynchronous state and localized in (c) and (d) for $\epsilon =0.9$ indicating CPS, respectively.

Figure 7

Confirmation of CPS in the chaotic regime using (a) four largest Lyapunov exponents and (b) correlation of probability of recurrence $\left(C_{CPR}\right)$.

Figure 8

Confirmation of CPS in the hyperchaotic regime using (a) ten largest Lyapunov exponents and (b) correlation of probability of recurrence $\left(C_{CPR}\right)$.

Figure 9

(Color online) CPS regime in the parameter space $\left(\epsilon ,\tau \right)$ characterized using the values of the index $C_{CPR}$.