Experimental synchronization of circuit oscillations induced by common telegraph noise

Experimental realization and quantitative investigation of common-noise-induced synchronization of limit-cycle oscillations subject to random telegraph signals are performed using an electronic oscillator circuit. Based on our previous formulation [K. Nagai et al., Phys. Rev. E 71, 036217 (2005)], dynamics of the circuit is described as random-phase mappings between two limit cycles. Lyapunov exponents characterizing the degree of synchronization are estimated from experimentally determined phase maps and compared with linear damping rates of phase differences measured directly. Noisy on-off intermittency of the phase difference as predicted by the theory is also confirmed experimentally.

Figure 1

(Color online) Experimental setup. (a) An electronic circuit used in the experiment. (b) A limit-cycle orbit observed at $V_g\left(t\right)\equiv -6.0\text{V}$ on the $\left(V_{+},V_{-}\right)$ plane. Average period was 0.0445 s. (c) Time series of $V_{+}\left(t\right)$ (left) and $V_{-}\left(t\right)$ (right).

Figure 2

(Color online) (a) Evolution of the absolute phase difference $\left|\Delta \varphi \left(t\right)\right|$ under a constant external signal $V_g\left(t\right)\equiv -6.0\text{V}$ depicted using doubly logarithmic scales. The solid curve represents the experimental data and the broken line with unit slope represents linear dependence on $t$. [(b) and (c)] Time series of the phase difference $\left|\Delta \varphi \left(t\right)\right|$ with $V_g\left(t\right)$ as a random telegraph signal ($V_{g1}=-6.0\text{V}$ and $V_{g2}=-2.5\text{V}$): (b) short-time series of $V_g\left(t\right)$ (top) and $\left|\Delta \varphi \left(t\right)\right|$ (bottom); (c) long-time series of $\left|\Delta \varphi \left(t\right)\right|$, exhibiting noisy on-off intermittency. (d) Distribution of the laminar length. The broken line represents $t^{-1.5}$. (e) Distribution of the absolute phase difference $\left|\Delta \varphi \left(t\right)\right|$. The broken line represents ${\left|\Delta \varphi \left(t\right)\right|}^{-1}$.

Figure 3

(Color online) Phase mappings between two limit cycles. (a) When $V_g\left(t\right)$ switches from $V_{g1}=-6.0\text{V}$ to $V_{g2}=-2.45\text{V}$, points on LC1 (solid curve) for $V_{g1}$ are mapped to the corresponding points on LC2 (broken curve) for $V_{g2}$ as indicated by arrows. (b) From $V_{g2}=-2.45\text{V}$ to $V_{g1}=-6.0\text{V}$.

Figure 4

(Color online) The probe signal used to determine the phase maps. Before $t=0$, $T_1$ and $T_2$ were measured. After $t=0$, the phases were measured at $t_1^i$ and $t_2^i$ for $1\le i\le 200$.

Figure 5

(Color online) (a) Experimentally determined phase maps. Circles show the raw data $\left({\Phi }_1^i,{\tilde{\Phi }}_2^i-{\Phi }_1^i\right)$ (left) or $\left({\Phi }_2^i,{\tilde{\Phi }}_1^i-{\Phi }_2^i\right)$ (right), and curves represent the nontrivial part of the estimated phase maps $f_{12}\left({\theta }_1\right)-{\theta }_1$ (left) or $f_{21}\left({\theta }_2\right)-{\theta }_2$ (right) obtained after low-pass filtering. $V_{g1}=-6.0\text{V}$ and $V_{g2}=-2.5\text{V}$. (b) Comparison of the Lyapunov exponents $\lambda ={\lambda }_1+{\lambda }_2$ with the linear damping rates $r_{\text{damp}}$ directly measured from $\left|\Delta \varphi \left(t\right)\right|$. The line represents $0.4r_{\text{damp}}={\lambda }_1+{\lambda }_2$. Filled circles represent the data obtained by varying $V_{g2}$ ($-2.55\text{V}\le V_{g2}\le -2.45\text{V}$ with intervals of 0.01 V) while fixing $V_{g1}$ at $-6.0\text{V}$.