Tuning the period of square-wave oscillations for delay-coupled optoelectronic systems

We analyze the response of two delay-coupled optoelectronic oscillators. Each oscillator operates under its own delayed feedback. We show that the system can display square-wave periodic solutions that can be synchronized in phase or out of phase depending on the ratio between self- and cross-delay times. Furthermore, we show that multiple periodic synchronized solutions can coexist for the same values of the fixed parameters. As a consequence, it is possible to generate square-wave oscillations with different periods by just changing the initial conditions.

Figure 1

Setup of two mutually coupled electro-optical delay systems with $T_f=T_{ii}$ and $T_c=T_{ji,j\ne i}$.

Figure 2

Hopf bifurcation curves for the in-phase (top) and antiphase (bottom) solutions with $n=1$ (solid red line), $n=2$ (dashed green line), and $n=3$ (dotted pink line).

Figure 3

Hopf bifurcation points appearing at ${\chi }_n=1$ as a function of $s_0$ leading to oscillatory solutions with different periods $T$. Squares and crosses correspond to in- and out-of-phase Hopf bifurcations, respectively. As $T$ approaches zero, the number of Hopf bifurcations increases dramatically (only the bifurcations verifying $T\ge 0.1$ are shown).

Figure 4

Numerical solution of the full dynamical model (9) for $\Phi =\pi /4$, $\gamma =0.5$, $P=1.7$, $T_f=30$ ns, and $T_c=90$ ns so that $s_0=1/3$, $\varepsilon =2.78\times {10}^{-4}$, and $\delta =0.018$.

Figure 5

Bifurcation diagram of the map (59) for $q=1$ and offset phases (a) $\Phi =\pi /4$, (b) $\Phi =0.3\pi$, and (c) $\Phi =0.35\pi$.

Figure 6

In-phase periodic oscillations after the Hopf bifurcation with $\Phi =\pi /4$, $\gamma =0.5$, $T_f=30$ ns, and $T_c=90$ ns so that $s_0=1/3$, $\varepsilon =2.78\times {10}^{-4}$, and $\delta =0.018$. In (a) and (b) we consider $P=1.01$ so that $\chi =1.01$, while in (c) and (d) $P=1.11$ ($\chi =1.11$).

Figure 7

Out-of-phase periodic oscillations after the Hopf bifurcation with $\Phi =\pi /4$, $\gamma =0.5$, $T_f=30$ ns, and $T_c=40$ ns so that $s_0=3/4$, $\varepsilon =6.25\times {10}^{-4}$, and $\delta =0.008$. In (a) and (b) we consider $P=1.01$ so that $\chi =1.01$, while in (c) and (d) $P=1.11$ ($\chi =1.11$).

Figure 8

Amplitude of the square-wave periodic oscillations as a function of the offset phase $\Phi$ with $P=1.11$ and $\gamma =0.5$. The solid line corresponds to the theoretical prediction. Symbols correspond to the numerical integration of the full dynamical model with $T_c=50$ ns and $T_f=60$ ns ($+$), $T_c=30$ ns and $T_f=40$ ns ($▵$), $T_c=30$ ns and $T_f=60$ ns ($\times$), $T_c=70$ ns and $T_f=50$ ns $⊲$, $T_c=30$ ns and $T_f=50$ ns $⊳$, and $T_c=30$ ns and $T_f=90$ ns $◯$.

Figure 9

Amplitude of the square-wave oscillations as a function of the offset phase $\Phi$ with $P=1.8$ and $\gamma =0.5$. The solid line corresponds to the theoretical prediction (59) with $q=1$. Symbols correspond to the numerical integration of the full dynamical model with $T_f=30$ ns and $T_c=90$ ns.

Figure 10

Time traces of the square-wave oscillations obtained for $P=1.8$, $\gamma =0.5$, $T_f=30$ ns, and $T_c=90$ ns for (a) and (b) $\Phi =0.15\pi$ and (c) and (d) $\Phi =0.18\pi$.

Figure 11

Time trace of the square-wave periodic solutions with $P=1.5$, $\gamma =0.5$, $\Phi =0.25\pi$, and $T_f=30$ ns. (a)–(d) Coexisting in-phase solutions for $T_c=90$ ns ($s_0=1/3$) obtained with suitable initial conditions as indicated in the text. (e)–(h) Coexisting out-of-phase solutions for $T_c=60$ ns ($s_0=1/2$). Also shown are (a) and (e) the fundamental solution $j=0$, (b) and (f) the first harmonic $j=1$, (c) and (g) the second harmonic $j=2$, and (d) and (h) the twentieth harmonic $j=20$. Notice that the time scale used in (d) and (h) is 10 times smaller than that in the other panels.

Figure 12

Dynamics of in-phase periodic solutions with $T_c=90$ ns and different values of $T_f$: (a) and (b) $T_f=30.0$ ns, (c) and (d) $T_f=29.6$ ns, (e) and (f) $T_f=29.4$ ns, and (g) and (h) $T_f=29.2$ ns.

Figure 13

Dynamics of out-of-phase periodic solutions with $T_c=40$ ns and different values of $T_f$: (a) and (b) $T_f=30.0$ ns, (c) and (d) $T_f=29.8$ ns, (e) and (f) $T_f=29.6$ ns, and (g) and (h) $T_f=29.4$ ns.