Two distinct bifurcation routes for delayed optoelectronic oscillators

We investigate the coexistence of low- and high-frequency oscillations in a delayed optoelectronic oscillator. We identify two nearby Hopf bifurcation points exhibiting low and high frequencies and demonstrate analytically how they lead to stable solutions. We then show numerically that these two branches of solutions undergo higher order instabilities as the feedback rate is increased but remain separated in the bifurcation diagram. The two bifurcation routes can be followed independently by either progressively increasing or decreasing the bifurcation parameter.

Figure 1

Schematic of the experimental setup.

Figure 2

The stability of the zero solution is bounded by the Hopf bifurcation lines $H_1$ and $H_2$ given by (3) and (4), respectively. Stable and unstable mean that the zero solution is either stable or unstable.

Figure 3

The two first Hopf bifurcations lines $H_1$ and $H_2$ in the $\beta$ versus $\delta$ stability diagram. They have been obtained by determining $\delta =\delta \left(\omega \right)$ from Eq. (14) and then $\beta$ using Eq. (13). The two Hopf bifurcation lines intersect at $\left(\delta ,\beta \right)=\left({\delta }^{*},{\beta }^{*}\right)=\left(0.146,1.079\right)$. The vertical broken line at $\delta =0.1411$ marks the value used in Ref. [34]. The values of the fixed parameters are $\varphi =\pi /4$ and $\gamma ={10}^{-2}.$ The first and second Hopf bifurcations are characterized by a frequency ${\omega }_1\ll 1$ and by a frequency ${\omega }_2\simeq 2\pi$, respectively. Stable and unstable mean that the steady state is stable and unstable, respectively.

Figure 4

Critical ${\delta }^{*}$ for a double Hopf bifurcation point as a function of $\gamma$. It has been obtained by first determining ${\omega }_1={\omega }_1\left(\gamma \right)$ from Eq. (20) and then by evaluating ${\delta }^{*}$ from Eq. (19). The vertical broken line denotes the value of $\gamma ={10}^{-2}$ used in Ref. [34].

Figure 5

Bifurcation diagram of the steady-state solutions of Eqs. (38) and (39). Panels (a) and (b) illustrate the case $c=-1$ and $c=1$, respectively. Both cases allow the coexistence of stable pure mode solutions through a secondary bifurcation mechanism.

Figure 6

Bifurcation diagrams of the extrema of $x$ obtained numerically from Eqs. (5) and (6) by progressively increasing (black) or decreasing (gray) $\beta$. The fixed parameters are $\varphi =\pi /4$, $\delta =0.1411$, and $\gamma ={10}^{-2}$.

Figure 7

Blow-up of the bifurcation diagrams near the two first Hopf bifurcations. Branch 1 (black dots) and Branch 2 (gray dots) correspond to the forward diagram of Fig. 6 (black line) and the backward diagram of Fig. 6 (gray line), respectively.