Simulink implementation of a new optoelectronic integrated circuit: stability analysis and infinite-scroll attractor

Abstract

In this paper a 6-D optoelectronic system consisting of an optical injected semiconductor laser driven by a resonant tunneling diode is reported. A stability analysis of the hybrid system is analytically and numerically performed and paramount role of the effective gain coefficient is stuck out in the framework of new stability control. As a result, this parameter allows improving the accuracy of the stability study by circumscribing locked and unlocked regions. Besides, a narrow area of stability is pointed up within the sea of unstable points from which a complex fractal attractor so-called infinite-scroll attractor is highligted. Thereby, Simulink shows generation effectiveness of infinite-scroll attractor erratically interpersed by laminar phases. Also dynamics of Lyapunov exponents has confirmed that it refers to a strange fractal attractor. Moreover chaos control is structurally carried out by direct current polarisation.

Appendix

The elements ${a}_{1}$, ${a}_{2}$, ${a}_{3}$, ${a}_{4}$ and ${a}_{5}$ of (4) and (5) are given by.

$$ a_{0} = 1,a_{1} = - 2\beta G_{0} \Delta n_{0} + mr + 3B_{0} m^{{ - 1}} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right) + \frac{{\gamma _{0} }}{{1 - \delta }}\left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - \delta } \right), $$

$$ \begin{aligned} a_{2} = & 1 + 3rB_{0} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right) + 2\beta G_{0} \gamma _{0} \left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - n_{0} } \right) + \left( {\alpha G_{0} \Delta n_{0} - \Delta \Omega } \right)^{2} \\ & {\text{ + }}\beta ^{2} (G_{0} )^{2} (\Delta n_{0} )^{2} - \left\{ {mr + 3B_{0} m^{{ - 1}} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right)} \right\}\left\{ {2\beta G_{0} \Delta n_{0} - \frac{{\gamma _{0} }}{{1 - \delta }}\left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - \delta } \right)} \right\} \\ & - 2\beta G_{0} \gamma _{0} \left( {\frac{{\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - \delta }}{{1 - \delta }}} \right)\Delta n_{0} \\ \end{aligned} $$

$$ \begin{aligned} a_{3} & = - 2\alpha G_{0} \gamma _{0} \left( {\alpha G_{0} \Delta n_{0} - \Delta \Omega } \right)\left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - n_{0} } \right) - 2\beta ^{2} (G_{0} )^{2} \gamma _{0} \left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - n_{0} } \right)\Delta n_{0} \\ & + \frac{{\gamma _{0} }}{{1 - \delta }}\left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - \delta } \right)\left[ {1 + 3rB_{0} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right) + \beta ^{2} (G_{0} )^{2} (\Delta n_{0} )^{2} + \left( {\alpha G_{0} \Delta n_{0} - \Delta \Omega } \right)^{2} } \right] \\ & + \left\{ {2\beta G_{0} \gamma _{0} \left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - n_{0} } \right) - 2\beta G_{0} \frac{{\gamma _{0} }}{{1 - \delta }}\left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - \delta } \right)\Delta n_{0} } \right. \\ & \left. { + \beta ^{2} (G_{0} )^{2} (\Delta n_{0} )^{2} + \left( {\alpha G_{0} \Delta n_{0} - \Delta \Omega } \right)^{2} } \right\} \times \left\{ {mr + 3B_{0} m^{{ - 1}} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right)} \right\} + \\ & - 2\beta G_{0} \left[ {1 + 3rB_{0} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right)} \right]\Delta n_{0} \\ \end{aligned} $$

$$ \begin{aligned} a_{4} = & \left\{ { - 2\alpha G_{0} \gamma _{0} \left( {\alpha G_{0} \Delta n_{0} - \Delta \Omega } \right)\left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - n_{0} } \right)} \right. \\ & - 2\beta ^{2} (G_{0} )^{2} \gamma _{0} \left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - n_{0} } \right)\Delta n_{0} + \\ & \left. { + \frac{{\gamma _{r} }}{{1 - \delta }}\left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - \delta } \right)\left[ {\beta ^{2} (G_{0} )^{2} (\Delta n_{0} )^{2} + \left( {\alpha G_{0} \Delta n_{0} - \Delta \Omega } \right)^{2} } \right]} \right\} \\ & \times \left\{ {mr + 3B_{0} m^{{ - 1}} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right)} \right\} \\ & + \left\{ {2\beta G_{0} \gamma _{0} \left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - n_{0} } \right)} \right. - 2\beta G_{0} \gamma _{0} \left( {\frac{{\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - \delta }}{{1 - \delta }}} \right)\Delta n_{0} \\ & {\text{ + }}\left. {\beta ^{2} (G_{0} )^{2} (\Delta n_{0} )^{2} + \left( {\alpha G_{0} \Delta n_{0} - \Delta \Omega } \right)^{2} } \right\} \times \left\{ {1 + 3rB_{0} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right)} \right\}, \\ \end{aligned} $$

$$ \begin{aligned} {\text{ }}a_{5} = & - 2\alpha G_{0} \gamma _{0} \left( {\alpha G_{0} \Delta n_{0} - \Delta \Omega } \right)\left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - n_{0} } \right)\left\{ {1 + 3rB_{0} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right)} \right\} \\ & - 2\beta ^{2} (G_{0} )^{2} \gamma _{0} \left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - n_{0} } \right)\left\{ {1 + 3rB_{0} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right)} \right\}\Delta n_{0} \\ & \frac{{\gamma _{0} }}{{1 - \delta }}\left( {\sigma B_{0} x_{0} (x_{0} - a)(x_{0} - b) - \delta } \right)\left[ {\beta ^{2} (G_{0} )^{2} (\Delta n_{0} )^{2} + \left( {\alpha G_{0} \Delta n_{0} - \Delta \Omega } \right)^{2} } \right]\left\{ {1 + 3rB_{0} \left( {3x_{0}^{2} - 2(a + b)x_{0} + ab} \right)} \right\}. \\ \end{aligned} $$

where ${x}_{0}$ and ${n}_{0}$ denote the steady states of the normalized electrical voltage and carrier density respectively.